Why do people keep sleeping on Burst Healing?

Mawtful

As already mentioned, the problem with Spider-Man is that Web Bandages is expensive AND limited in scope. That's like getting the worst of both worlds.

And then there's the meta-game.
MPQ rewards fast wins. Primarily, finishing a match quickly means that you're back in a position where you can purchase a shield if necessary. Shields preserve your points and you need to do that if you want to hit the top rewards in PvP. The other great benefit to a quick win is your opponent can't damage you if they're already KO'd. Web Bandages is, quite literally, one of the worst abilities in the meta.

Medical Marvel is not much further ahead - even at rank 5 the protect tiles it creates are weaker than any protect tiles that Mutagenic Breakthrough might create, and you need a Blue special tile in play to even get those Protect tiles from Medical Marvel. However Beast actually provides some options or range to his utility. Mutagenic Breakthrough is fast and can provide strike or attack tiles, and his AoE has decent damage:ap ratio if there are special tiles in play. Beast's biggest problem is living in the shadow of Rocket & Groot, whose Blue and Green abilities outclass Beast entirely.

And then we have the head of the class, Kamala Khan. It would be easy to call her healing "free" and be done with it. I think a better way to really **** its usefulness instead of viewing her passive as providing healing when an ally uses an ability (this is how the ability is written and mechanically how the ability works) but rather to view her ability as providing burst healing to each of her allies' abilities. For 5 AP, Gamora can deal damage to a target enemy and heals the whole team; for 8 AP, Psylocke deals damage, creates a Strike tile and heals the whole team; etc. A lot of abilities increase in value significantly when they also provide a nice burst of healing. What's more, Kamala has a cheap AP acceleration which can serve double duty in denying a non-Green colour to your opponent AND she's got arguably the best AoE in the 3* tier (Panther is slightly stronger, but could give away important AP; Kamala has no draw back). In the interest of fairness, I'll also assess Spider-Man's utility which he provides in the form of a cheap stun. It's 1, 2, or 3 turns depending on how many web tiles you have, and you could achieve the same or better with a 1* Black Widow team-up which would cost less and let you take someone else on your team who can end the match faster than Spidey.

So, what do we do with Burst Healing? Get rid of it entirely? Maybe.
In my opinion, a Burst Healing ability will only be considered viable when it also does something else - the "something else" will be the reason why people want that ability and the burst healing will be the sweetener which seals the deal. So, Beast is a great example where the "something else" isn't worth bringing (11 AP for two weak protect tiles if and only if you've got friendly Blue special tiles in play), whereas Kamala highlights abilities which may only be "average" being considered as "good" or better because of the bonus healing. I think the biggest problem, from a design point of view, is that Demiurge is still hung up on the idea that healing is pacifist or defensive concept. Burst Healing is almost perfectly primed to represent energy transfer/generation or even life-drain abilities. Ragnarok's Blue ability touches on this as it Charges Green tiles and provides him a burst of health. Now I think the Ragnarok redesign is a great example of how not to design, this ability is at least a peek at a different way of looking at Burst Healing.
The other consideration that needs to be made is the temporary nature of Burst Healing. It's only there for the duration of the fight, so I think:
it needs to heal for bigger numbers, and/or
it needs to act like a "shield" and allow over-healing, and/or
it needs to be able to bring a KO'd ally back into the fight

Based on the developer's description/explanation of Burst Healing (that is, being a "second wind" to keep characters going for longer) there's no reason why it shouldn't be able to 'revive' KO'd characters. That alone would probably make me consider Spidey on a few different teams.

LegendReborn

As others have said, Spidey's healing suffers from a lot of things like it being too expensive, it not healing enough for its cost and also removing web tiles.

If the developers really want to make Spidey's yellow half decent, they just need to cut the cost of it in half, reduce the healing by 1/2. They could have an optional part of the power consume web tiles for an extra burst of healing but having it be a required part of the ability totally kills it. Personally, I think the consumption of web tiles should add something like damage so spidey brings more than protection.

BAM! Spidey's healing is now not only viable but actually works with his kit rather than against it.