I think that self-sacrificing moves can be very powerful and engaging gameplay features as well as story features if developed enough, but most people tend to avoid them be just smacking people with Attack is seen as better than hurting yourself just to do a little more damage. It's probably best to make it an optional skill tree or party member that the player can opt into if they wish rather than something forced on them. But it can be loads of fun and I've seen it done excellently in 2 games:
Final Fantasy IV: This is without a doubt the mastercraft example. The main character, Cecil, actually begins the game as a Dark Knight in service to a kingdom that has him do some pretty amoral things that start to weigh on him. Fitting his starting role, he gets an ability which sacrifices some of his health to nuke the entire enemy side of the field. This is just plain amazing early game and most players will probably use it for some easy grinding to help out Rydia - a mage character who starts out very low level - since she'll be your only party member for a bit. The game is pretty good at making you actually use the skill and rewarding you well for it. But the real payoff is later in the game. Having resolved to mend his ways and stand up against his own kingdom, Cecil goes to the Mountain of Ordeals where he can class change. But the trick here is that all of the enemies on the mountain are undead...so they basically negate his self-sacrifice ability which is dark elemental type. Then Cecil has to fight a solo boss fight against a dark version of himself that represents his inner darkness. Naturally, his inner darkness spams his self-sacrifice ability every turn to nuke you down. But the catch is that in the way to win this battle is actually to defend instead of attack. If you fight fire with fire, then you just burn down your HP even faster. But if you take the high road and focus on defense, then you can keep your HP up while your opponent kills himself. The fight completes Cecil's main character arc by showing both Cecil and the player that true self-destructive nature of "the dark side" and allows Cecil to class change into a Paladin, losing his self-sacrifice skill in exchange for healing magic and being able to take attacks for his allies.
Magic the Gathering: This is definitely the pure strategic choice of example. Being a tabletop card game, the numbers of cards you have access to is an important element of the game, and having more cards in your hand than your opponent is very good tactically. Cards are used to deal damage to your opponent's Life total while protecting your own. Then along came Necropotence. It has 3 effects: 1) Skip your own draw step 2) If you discard a card from your hand, remove that card from the game 3) Pay 1 Life to remove the top card of your deck from the game and add it to your hand you next discard phase (which is the end of your turn). On the surface this card looks utterly terrible - you can't draw cards like normal and have to pay life to draw anything, and you can't discard cards so you can't even combo it with cards that let you bring things back from the Graveyard, and on top if it all you don't even actually "draw" the cards so you don't get them instantly. And yet this card completely changed the entire metagame of Magic. It's self-sacrificing, but by changing your mindset you can get an enormous amount of power for that sacrifice. The extra cards you get cost no mana, meaning that 100% of your mana can go into casting spells that directly hurt your opponent and/or heal yourself instead of paying to draw extra cards. You can use the effect to have a full hand every turn, draw enough cards each turn to always be able to play a Land card, and always have more mana and more available spells than your opponent. And since you skip your draw step you don't lose the game if you run out of cards in your deck, since you never have to draw from it. It completely changes how you have to play the game, but if you base your play-style around it then the sacrifice becomes an awesome advantage. Necropotence completely changed how even the game's designers looked at the game - instead of every Life Point being a precious thing that brought you closer to defeat as it went down, it became another resource to be spent to ensure your victory. Because the only Life Point that matters is your last.
There is a ton of tactical decisions that can be made when you can use your own HP as a resource, but the payoff has to be really good - either in magnitude or just effects that you can't get any other way. If you base a combat system on it or even just one character, then it can offer a lot of fun gameplay for people that don't mind taking risks. But just throwing it into a combat system not designed for that kind of risk/reward pattern can just feel superfluous and is probably why many shy away from them when they see them.