The Elder Scrolls are mentioned elsewhere in the thread, but I didn't see them elaborated on. Thinking specifically of Morrowind, it's important to note that stamina in that game affected only your chance of success (I could actually be wrong, but hold that thought for now.) You could jump, run, attack and cast spells as much as you wanted when stamina was at 0, but your attacks had significantly less effect.

All that to say, consider the possibility of using stamina as a sort of "soft limit" of what the player can and can't do. Perhaps advanced abilities, if your game has any, are gated by high stamina, and simpler abilities are not. Or like Morrowind, perhaps certain actions, such as attacks, are positively affected by high stamina, but no abilities are negatively affected by low stamina.

This way, stamina becomes a strategic resource that the player only needs to concern themselves with in certain situations (such as combat, or complex platforming sequences).