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In Magic, the player who goes first doesn't get to draw a card on the first turn, i.e. the player who goes second plays each turn with an extra card. This typically isn't enough compensation and virtually everyone chooses to go first in constructed.

In addition to this mechanic, Hearthstone also allows the second player to start with a "Coin", which is effectively Lotus Petal. Can Magic adopt this mechanic as well?

Obviously Magic is more complicated than Hearthstone and Lotus Petal introduces new risks, e.g. Lotus Petal is a permanent (e.g. counts for City's Blessing), an artifact (e.g. counts for Improvise), can be sacrificed (e.g. counts for Revolt), and so on. However, I can't think of any way this lets the second player do really degenerate things like win on turn 1. Instead, this would let the second player counter the first player's turn 1 win attempts by e.g. casting Spell Pierce. Conversely the first player is already able to interact with the second player with cards like Thoughtseize before the second player ever reaches a main phase.

Is there any dangerous combo that could be enabled by letting the second player start with a Lotus Petal in play? If so, what combo could it be?

There are already decks that win on turn one, giving one extra mana on that turn enables these decks to be more reliable.

One example of a deck that wins turn one, and already wants to be on the draw, is Flash Hulk. While Flash has been banned in legacy and restricted in vintage, the combo only needs two mana, one of it blue, to win turn 1. This two mana is usually accomplished with Gemstone Caverns, Elvish Spirit Guide or Simian Spirit Guide and the land drop for the turn, usually an Island. Adding one free mana would mean you no longer need one of those three cards, just Flash and Protean Hulk and your land for turn in the opening hand.

Flash is used to put Protean Hulk into play, which must be sacrificed since you can't pay the 4 generic mana required by Flash (creature's CMC-2), triggering Protean Hulk's death effect. That effect could search for many things, 4 Disciple of the Vault and X cost artifact creatures like Phyrexian Marauder and Shifting Wall, using the 0 toughness artifact creatures dying to trigger the Disciples, or 4 Virulent Sliver and a Heart Sliver to swing 5 poisonous 4 hasted creatures turn 1, doing double the lethal poison count.

Mana and land is an integral part of the Magic. It controls the pacing of the game and gives you a limited resource which can make for interesting choices. A reason the one land per turn rule exists is to enforce that pacing and make sure both sides have a fair chance to build up and respond to an opponent's threats.

Many decks build their mana base so that they can hit certain drops as early as possible. If your deck runs on 3 mana (most of your key cards cost 3), you make sure you can reliably drop a land every turn for the first 3 turns. If your deck needs to hit 4 drops on turn 4, you add more mana or add ramp. If your deck has multiple colors, you think about adding color fixing so you can play what you need to as early as possible.

Think about what happens when you miss a crucial land drop. Your turn is now probably wasted and you are put at a great disadvantage. Each increase in cost generally coincides with an increase in power. It's hard to keep pace with an opponent's 3 drops when you can only play 2 cost spells.

So what happens if you gave one person a free Lotus Petal? Well, you've just given that player a turn advantage (they can play a spell a turn ahead now), ramp and color fixing. That's powerful. And that doesn't even include the synergies you've touched on.

Imagine what you could do if you had one extra mana, even if it was just once. Turn 2 Steel Leaf Champion? How would you respond to that? Turn 3 Liliana of the Dark Realms? Yes please. If you are playing burn, how awesome would it be to get to drop an extra burn spell early? Playing aggro? Establish your board dominance faster.

The single extra mana alone would probably be too powerful, let alone any synergies from having a bonus sacable artifact on the field.

Practical implementation problems of paper magic aside, the effect of the change would depend on the format. The faster the format, the more likely the change would warp it. I would expect a free Petal for the player to strongly warp the game in favor of going second. Considering that Petal has in the past been banned/restricted from competitive play for some years and has only been reprinted for online play, the chances of WotC giving everyone free a Lotus Petal on their starting hand for being on the draw are slim to none.

From a practical standpoint, in paper Magic you would have to provide and distribute what amounts to Lotus Petal token cards from now to until the rule is abolished. Since it's a general rule, you could not limit the distribution to certain blocks, like with transform proxy cards. That limits how many other token cards you could include in future sets.

You would sharply devalue the secondary market value of actual Lotus Petals, both paper and online, because there would be much less incentive to include them in your deck. While Lotus Petal is obviously nowhere near power 9 levels of prices, WotC has been known to respect the secondary market by vowing to never reprint certain rare cards on paper. Considering the similar gameplay purpose of Lotus Petal, the change might even negatively affect prices of Moxes and Lotus.

From a gameplay perspective, the Petal would have the biggest effect in the eternal formats, where a match can be decided within few turns. As you already mentioned, there is a multitude of archetypes that would be affected by a free (as in hand cards), free (as in mana cost) artifact that produces mana. It would most likely promote combo decks that need to generate a certain amount of mana within a single turn to go off.

I can't go into specific combos that would be enabled by Lotus Petal. Suffice to say that there are already decks that win on turn 1, and a Petal would only enable more of them. That is in itself not necessarily a bad thing, the meta would adapt around that.

In any case, constructing and piloting a deck would become much more complex, because being on the play and on the draw would be even more different than today. In a way, it would also make the outcome of games, at least the first in a series, more luck-dependent. Since a Petal is such a strong card, most players will be compelled to optimize their deck for having one, and that means you are more dependant on the luck of the coin flip in the first match of a series. That can't be healthy for a game that already involves quite a bit of RNG.

It makes the game more complex

This might seem like a minor point for a game as complex as Magic is. But this particular change introduces several different types of complexity in a way that is unhealthy for the game.

In order for Magic to stay healthy it requires a constant influx of new players. Magic has too many moving pieces for most new players to absorb it whole, so it's vital that new players be introduced to this complexity a little bit at a time.

Fortunately, Magic lends itself well to this sort of thing because almost all of the complexity is in card interactions, so players are naturally eased into the game as the pool of cards they encounter grows.

This free Lotus Petal that you propose violates that. It drops a one use only mana source onto the board at the start of every game. As game objects go it's no Ice Cauldron, but no Grizzly Bears either, and the very first object that a new player encounters in the game needs to be as simple as possible. (For example, right now, the first object most new players encounter is a basic land)

In 2011, Mark Rosewater wrote an article called New World Order

In it, he talks about how R&D came to realize that complexity was being introduced to new players too quickly.

He describes three types of complexity: comprehension complexity (how hard it is to understand what a card does), board complexity (how hard it is to understand the current state of the board), and strategic complexity (how hard it is to understand what you should do next).

The free Lotus Petal adds very little board complexity, and its comprehension complexity isn't particularly high either (although considering how many new players don't understand the relationship between mana and lands, I suspect it's still high enough to cause problems.) The big problem is strategic complexity.

For a new player with a simple deck, early strategy is simple. Choose to go first (if given the choice), play a land every turn, then play the most expensive card you can (unless it's obviously not useful). A new player with a simple deck can make it half a dozen turns into a game before needing to make a decision that isn't blindingly obvious.*

The Lotus Petal changes that. Do you play first or second? answer: It probably depends on what kind of deck you are playing. Do you pop the Lotus to pay for a 3 CMC card on turn 2, or a 5 CMC card on turn 4? answer: That depends on what you have in your hand.

A new player isn't equipped to answer these questions, and forcing them to have to deal with them at the beginning of every game is only going to frustrate them.

Even if the Lotus Petal improved the first/second player game balance, the added complexity it adds would harm the game more than the slight advantage first player gets.

*Savvy readers will notice that I left out the mulligan rule, which is technically the first complicated decision a player must make. But even that has pretty easy guidelines (3-4 lands or mulligan), and because one of the mulligan choices is "do nothing" it can be left out of early games entirely without significantly harming the teaching experience. The same is not true of the Lotus Petal which would need to be included in from the very beginning.

That would not work because the imbalance is much smaller than you imagine it is.

The only proof of imbalance I could find is this document which shows the first player has a 53% chance of winning instead of 50%. Balancing the game would require giving the second player an extra three wins every 100 games.

The change you suggest is extremely powerful. It allows to player to get a larger drop or more drops much sooner. It allows responses to be threatened much sooner. It allows color fixing in a multi-color deck. Or any number of other great advantages.

The change would therefore result in much much much more than a mere 3 more wins per 100 games for the second player, and is therefore broken.