"Since UUs boosting useless units stay useless"
Or you could make those units less useless.
Why have useless units at all?
"We need to find more good basic units for UUs."
Agreed on this, however.
"England would qualify for Rifles or Line Infantries ('Redcoats'), but that wouldn't contribute to diversity."
Actually Man-o-War is a nice UU with its 4 RoF. I've used it extensively for city bombardment (more mobile than contemporary arty).
"Then, we shouldn't try to be different just for the sake of being different, so Gallics, Legionaries, Nummies, Hoplites, Riders should stay."
Nothing is so good it can't stand further improvement, but these are good enough that as to be low priority.
"For Mali and Arabs, Knight replacements are fitting extremely well. Same for Mongols and Sioux with Horse Archers."
I would disagree here. Both the Heavy Horse and the MW are boring compared to, say, the Cossack. Need more uniqueness.
"Now, if someone could come up with a better UU for Siam, I'm open for any suggestions."
I actually find the Siam UU more interesting, say, than the Sioux, in that it has uniquely high attack power for its era. Similar to the unmodded Immortal. Though that doesn't last terribly long.
"And maybe we could move the War Elephant to Elephant Training; this is an AA tech, but you can get to Knights at the same time."
Here's the kind of thing I have in mind: American UU -> Minuteman, Rifle, 5/5/1 ATAR, defensive bombard 10, 2 RoF. This gives you the 5 def you're looking for from a Rifle, with significantly improved mobility and a unique feature (increased RoF for defensive bombard). You give up 2 on the attack (implying a nice combined arms strategy). Might also offer slightly decreased shield cost. edit: looking at the tech tree, this is anachronistic - would either need to be a line infantry variant (with nationalism), or have a civil war-era name.
Considering your list:
1 Champ (Aztecs)
2 Archers (Babylon, Iros)
3 Spears (Greece, Carth, Ethopia)
1 Chariot (Egypt)
3 Swords (Rome, Celts, Siam)
2 Horse Archers (Mongols, Sioux)
1 Pike (Tibet)
2 MDI (Inca, Vikings)
4 Knights (Mali, Arabs, China, India)
1 Cavalier (Japan)
1 Explorer (Spain)
1 Cannon (Korea)
1 Arquebusier (France)
1 Cuir (Ottomans)
1 Dragoon (Russia)
1 Ship-of-the-Line (England)
1 Para (US)
1 Armor (Germany)
Babs, Ethiopia, Sioux, Siam, Vikes, Mali, India, US, and Germany could move to neglected units (Modern UU's are a bad idea as thier impact is negligible in most games). You could use a Horse UU for Sioux, Camel for Ethiopia (don't need to require the resource - see jumbos in unmodded), Elephant for India, Hussar for Germany, Rifle for US (see above), Xbow? for Vikes, Cataphract for Babs (no one would mourn the loss of the bleh Bow), etc...
Also wouldn't rule out unique versions of Setlers and Workers, as well as boats that can do special things (like enter ocean a little early - lot of room for play there).
I hadn't realized how much variety you already had, BTW. good work.