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Currently, enchantments cause the game's balance to be quite questionable, resulting in stuff such as:

• Arrows being almost pointless to use on crossbows (Why would you? The bow will fire faster and 1-hit anything it kills. It's quite likely you'll actually save up arrows and deal more damage using it on crowds over a Piercing crossbow!).
• Sharpness being practically the end-all-be-all of sword damage enchantments (BoA makes you deal more damage on already-low-health mobs, and Smite is only good against Wither, but Sharpness is always useful).
• Protection being omnipresent (Why have better protection against ONE thing when you can have good protection against pretty much everything?)

Now, I consider that BoA and Smite are fine on their own, and the real problem is that there's no real point to a specialized enchantment when there's a general, mutually exclusive one with the same effect but somewhat weaker in most cases. Now, with a few tweaks and changes the problem could more or less go away:
Changed enchantments:

1. Sharpness: Rather than providing a general damage increase, it'd become specialized the same way Smite and BoA are, making each damage enchantment more useful due to not having to compete with a general enchantment (and losing). Sharpness' effect is now the following: It damages any mob NOT covered by other damaging enchantments. This means that it doesn't work on arthropods, undead, or otherworldy enemies (See new enchantments), but affects mundane, living mobs that don't have an exoskeleton.
2. Power: Rather than just giving more damage, it has a different effect: It allows the Player to draw the bow further back by 1 stage, increasing the intensity of the enchantment glint each time. Each additional charge stage deals just as much damage as a fully-charged bow now deals with the corresponding Power level. Each stage adds +0.3 seconds to fully charge, althrough an enchanted bow can still be timed to make it shoot just as fast and deal just as much damage as a normal bow (In other words, Power lets you overcharge it for more damage).
3. Protection: Now only protects against melee damage and, to a lesser extent, fall damage as opposed to protecting against everything.

New enchantments:

• Banish: Sword enchantment that increases damage dealt to "eldritch" mobs: Endermen, Ender Dragon, Blaze, Shulker, Guardian, Warden (Likely). Essentially, if it's weird and has an uncanny combat ability, it can be expected to be affected by Banish.

Enchantment Tweaks:

• Rather than one sole damaging enchantment or protective enchantment being allowed in a given sword or armor piece, they'd have a new mechanic: A single piece of gear can have up to 2 of these enchantments at once, so as to allow for more versatile gear (Specially now that there's no true general-purpose enchantment). This means that, for instance, a sword can have BoA and Sharpness at once, but not BoA, Sharpness and Smite all at once.
• Some mobs can fall under several categories where it comes to damaging enchantments: For instance, the Wither becomes affected both by Smite and Banish, as do Phantoms. Endermites could also be affected by both Banish and Bane of Arthropods. This also allows for future mobs to fall under these categories according to design (For instance, a hypothetical mob that's essentially a Cordyceps-infested ant could be affected by both Smite and Bane of Arthropods, and an Illager that has ended up mutated due to dark magic could easily be damaged by both Banish and Sharpness).

You're absolutely correct with your evaluation of enchantments, and I think these are pretty good ways to solve these issues without being too much work

That being said, another option is to go all in with the workload, and to overhaul these enchantments.

Which I will do to amuse myself haha.

Rather than the 'flat damage' enchantments being focused on dealing damage to specific enemies, they could deal damage based off certain conditions. Here's some ideas:

Chain strike - after defeating an enemy, you deal more damage for X seconds (X being the level of enchantment, does not stack with itself)

Falling strike - increases damage with jump attacks

Speed strike - your cooldowns/attack animations are shorter

Welp.

I'm bored now...

Let's just stick with your idea, thinking is hard.

Quote from AMPPL50»

Currently, enchantments cause the game's balance to be quite questionable, resulting in stuff such as:

• Arrows being almost pointless to use on crossbows (Why would you? The bow will fire faster and 1-hit anything it kills. It's quite likely you'll actually save up arrows and deal more damage using it on crowds over a Piercing crossbow!).
• Sharpness being practically the end-all-be-all of sword damage enchantments (BoA makes you deal more damage on already-low-health mobs, and Smite is only good against Wither, but Sharpness is always useful).
• Protection being omnipresent (Why have better protection against ONE thing when you can have good protection against pretty much everything?)

Now, I consider that BoA and Smite are fine on their own, and the real problem is that there's no real point to a specialized enchantment when there's a general, mutually exclusive one with the same effect but somewhat weaker in most cases. Now, with a few tweaks and changes the problem could more or less go away:
Changed enchantments:

1. Sharpness: Rather than providing a general damage increase, it'd become specialized the same way Smite and BoA are, making each damage enchantment more useful due to not having to compete with a general enchantment (and losing). Sharpness' effect is now the following: It damages any mob NOT covered by other damaging enchantments. This means that it doesn't work on arthropods, undead, or otherworldy enemies (See new enchantments), but affects mundane, living mobs that don't have an exoskeleton.
2. Power: Rather than just giving more damage, it has a different effect: It allows the Player to draw the bow further back by 1 stage, increasing the intensity of the enchantment glint each time. Each additional charge stage deals just as much damage as a fully-charged bow now deals with the corresponding Power level. Each stage adds +0.3 seconds to fully charge, althrough an enchanted bow can still be timed to make it shoot just as fast and deal just as much damage as a normal bow (In other words, Power lets you overcharge it for more damage).
3. Protection: Now only protects against melee damage and, to a lesser extent, fall damage as opposed to protecting against everything.

New enchantments:

• Banish: Sword enchantment that increases damage dealt to "eldritch" mobs: Endermen, Ender Dragon, Blaze, Shulker, Guardian, Warden (Likely). Essentially, if it's weird and has an uncanny combat ability, it can be expected to be affected by Banish.

Enchantment Tweaks:

• Rather than one sole damaging enchantment or protective enchantment being allowed in a given sword or armor piece, they'd have a new mechanic: A single piece of gear can have up to 2 of these enchantments at once, so as to allow for more versatile gear (Specially now that there's no true general-purpose enchantment). This means that, for instance, a sword can have BoA and Sharpness at once, but not BoA, Sharpness and Smite all at once.
• Some mobs can fall under several categories where it comes to damaging enchantments: For instance, the Wither becomes affected both by Smite and Banish, as do Phantoms. Endermites could also be affected by both Banish and Bane of Arthropods. This also allows for future mobs to fall under these categories according to design (For instance, a hypothetical mob that's essentially a Cordyceps-infested ant could be affected by both Smite and Bane of Arthropods, and an Illager that has ended up mutated due to dark magic could easily be damaged by both Banish and Sharpness).

Prot I-IV is like the best protection enchantment and the worst prot enchantment is projectile.

Sharp is the best sword enchant other than mending BoA is just the worst out of sharp and smite and the worst sword enchant is fire aspect because for instance kill an endermen if you hit it with fire aspect it will teleport until the fire is gone and you will not see it again.

and banish does not sound like a good enchant because why would you need it it makes no sense to have it at all.

Rather than making generalist Power, Protection and Sharpness specialised, I'd add "melee protection" and "bleeding" enchantments as extras and make the generalist enchantments either weaker, or give them specific drawbacks in other properties, like consuming x3 as many durability points, slower striking rate, slowing side and reverse movement or making bow draw arrows slower.

Banish does make plenty of sense if Sharpness becomes specialized: For starters, the mobs it includes tend to be powerful and tanky, with one being a boss outright.