We've anyway seen new textures already, so where is the love for the animations?
Update Aquatic saw some cool animations: activated conduit, riptide, loyalty, turtles laying eggs, sneaking and the dynamic sea grass and kelp.
These animations made the aquatic update that much cooler, but, I think we can go further.
We could add in so much more animations to the game! For example, more dynamic blocks (soul sand faces could open and close their mouths repeatedly, jack o' lanterns could have a 'pulsing glow' due to the torch - a similar effect could be brought to torches Beacon beam doesn't just randomly appear, it should have a nice animation to show the beam rising towards the sky. Redstone in general should have little sparks of redstone flying off, but only if activated), Villagers (and maybe some other mobs) should have facial expressions for agreeing, disagreeing, getting hurt and for dying.
Sword glinting during the day (Reflective surface, not because of enchants)
Leaves cosmetically falling from leave blocks to the ground. (Won't affect gameplay).
Real knees, elbows and fingers/toes.
Better death animation. Instead of just falling over and then just 'poofing' away. Player should have multiple animations, for example, weak in the knees and then slowly going down until falling on face or back.
These animations would really spice up Minecraft.
When a horse, mule or donkey moves, mane and tail should have a flowing effect, similar to real horses.
When creeper explodes, little creeper pieces should fly outwards (cosmetic only.).
Moving doors, trapdoors and fence gates, instead of from open to close and back instantly.
Elytra flight animations, (Various poses, Particle effect coming off elytra similar to the effect from a Phantom's wings.).
Better animation for squids swimming.
Loading animation added to the loading screen.
When players start to drown, they drop items in their hands and begin to clutch their throat. (A bit scary, I know...)
When players sleep, they pull bed sheets on to themselves and sleep in various different poses.
Edit: Going off of what Isacplaner said about Mojang/Microsoft losing their brand, maybe this could be done in a way where, you can complete specific advancements and unlock a random animation?
As a side note, swimming isn't really an animation, it's more of a mechanic IMO.
Animation would make for a more aesthetically pleasing world. It doesn't make sense that swimming has no animation but I'm afraid it might lag the game with a large amount of them going. Only a select few should have animation to avoid this problem.
New animations wouldn't create lag no matter how many you have going on. There would be the same amount of lag as there was before.
It won't happen though for multiple reasons.
1. The player model is rigid.
Steve has no joints. You can't bend him and have it look natural. While you may scream out and cite your Minecraft kinematics and mods, no matter how much work you put into it you won't be able to achieve the polish which Mojang so desperately attempts to achieve with every iteration of Minecraft.
2. It just isn't minecraft.
Mojang knows their brand. They know that if they change (much of) the graphics, that they'd lose a major tool at their disposal, which is brand recognition.