Endermites are nearly useless in current Minecraft. They can only spawn from thrown Ender Pearls with only a 1/20 chance. Their only drop is three XP orbs. Their only worth comes from their rarity, making them great pets, and their Endermen attraction, allowing them to be used in Endermen farms. It's safe to say many players aren't even aware they exist.
I'd say Endermites can be improved in three ways: improve their spawning, make it clear they can come from Ender Pearl throws, and give players a reason to seek them out more than once or twice.
Advancements serve as a natural path of progression through the world. The natural progression of advancements commands players to build a Nether Portal before ordering Eyes of Ender on the search for a stronghold. However, advancements also serve as mini-tutorials to introduce game mechanics, such as 1.16's upcoming advancements "Oh Shiny," which teaches players that dropping gold allows them to barter with piglins, and "This Boat Has Legs," which teaches them that Striders can be ridden across lava by holding a Warped Fungus on a Stick.
As such, it would be fitting to create an advancement teaching players that they can spawn Endermites by throwing Ender Pearls. This advancement could be called "Infestation" since Endermites are considered arthropods and are similar to silverfish, which are considered pests. It would make sense for Infestation to branch off from Remote Gateway since that advancement teaches players how to throw Ender Pearls.
As for improving Endermite spawning, the best option is adding them to the Outer End Islands as semi-rare spawns. (Think natural Witch rarity for reference.) That way, players would be able to find them naturally, and the Outer End will become slightly more dangerous. These Endermites would have a few differences between them and Pearl-spawned ones. Namely, Endermen would not be aggressive towards these natural Endermites for two reasons. One, the Outer End would be just a little more dangerous with two types of mobs ready to attack the player. Two, Endermites have very little health, so Endermen naturally attacking them would result in them all dying before players could find them.
In addition, Outer End Endermites would have a 20% speed boost, equivalent to the Speed 1 potion effect, to help mitigate their squishiness from their eight health. Sure, most players would likely visit the Outer End with at least a Sharpness diamond sword, so they could one-shot Endermites regardless. But they'd have to work a little harder aiming, at least. Or they could use a bow. It's not like the Outer End has anything in terms of difficulty aside from End Cities and the gaps between islands.
And finally, Endermites receive a drop called the Mite Limb. It looks like a purple stick with another small purple section branching off near the top like typical sticks are pictured. (I can't make a visual myself, so if anyone wants to, feel free.) The Mite Limb will only drop from player or wolf kills like Blaze Rods. An Endermite has a 50% chance to drop a Mite Limb, with an extra 10% per level of Looting, maxing out with an 80% chance to drop at Looting 3.
Mite Limbs are chargeable, throwable weapons. They charge as fast as Tridents but can be thrown at any point during the charging sequence. They deal half a heart when under 25% charge, gaining another half a heart per quarter of charge. They max out at two hearts of damage and have significantly less gravity pull when thrown at full charge. However, they also shatter upon impacting anything (with one exception), so they are non-reusable. Their special effect is unique. Thrown Mite Limbs pierce any Protection enchantments or Resistance effects. This seems powerful at first, but a full set of diamond or netherite will reduce this to just about half a heart of damage regardless of magical protection. According to the Wiki, even a full set of Iron armor would reduce this value to one full heart. Regardless, they only drop from player/wolf kills and aren't a guaranteed drop no matter what. They also have a reduced stack size like other throwables, maxing out at 16. This could possibly be reduced further to 8. Then there's also the shattering downside.
The main use of Mite Limbs is against Shulkers. A fully-charged Mite Limb pierces a Shulker's shell by halving its effective armor when dealing damage, allowing it to do either two or three damage points depending on how rounding works. The limb also sticks, forcing the Shulker to exit block mode immediately, and it can't return until another attack shatters the limb. The main reasoning is to deal with those annoying Shulkers that sit in block mode on the bottom of End Cities. That way, the player can snipe them from a distance to chip away at them rather than having to climb, fly or levitate up to them.