I still haven't gotten new controller & will probably not acquire new one any time soon.
I have tried following games with my controller:

Iconoclast - playable, but not quite there
Semblance - didn't know what to do with it
Trine - awful experience, wizard is quite unplayble there
Nine Parchments - I would rather play this one with kb+m, much better experience
Dead Cells - after playing with kb+m I felt lost

Which games should I try to learn using controller?
Games that I own, that are playble with controller

full Trine series
Shadwen
Regalia - on steam
A Hat in Time
Haimrik
Moon Hunters
Divinity OS 1&2
CrossCode
Battle Chef Brigade
Bard's Tale 4 - maybe?
AER - Memories of Old
Aegis Defender
Tangledeep
Ghost of a Tale
Kynseed
Ori and the Blind Forest
Mulaka
Witcher 3 GOTY
and many more

Divinity works surprisingly well with controller and since its turn based you don't really have a lot of hectic moments that just don't work if you're not used to controllers.

Shadwen plays nice too, and I actually preferred to play Witcher III with controller. Indoor-Gerald is still in tank mode, but outdoor movement and fighting feels very fluid without a need to be too precise with the analogue sticks.

Definitely play this (excellent) game with a controller. I played it with a XBOX360 gamepad and it was a blast. I cannot even imagine playing this with keyboard & mouse. I cannot comment on the rest of the games you mentioned, though, as I haven't played any of those.

I would suggest looking at genres that have historicaly been popular on video game consoles, like 2D platformers. From your list, you might try Battle Chef Brigade or Ori and the Blind Forest.

Games I've played that work well with a controller include:
* Timespinner
* Hollow Knight (from what I've played)
* Shovel Knight
* Wonder Boy: The Dragon's Trap

Tangledeep is an interesting question. The one problem I have with console roguelikes (like Shiren the Wanderer) is diagonal movement; if the game does not have that, then there's no reason that game shouldn't be playable. (Tanglewood, on the other hand, was designed with a Sega Genesis style controller in mind, but that's a completely different game.)

The Ys games also play decently with a controller.

Depends on what problems you want to overcome with the controller. If it's just general playing/working with, try slow games like Turn Based Divinity OS or stealthy sneaky Ghost of a Tale.

The Witcher 3 is great to play with a controller. The way you control that game is used in many other modern games as well.

Play "Brothers: A Tale of Two Sons" with a controller.
Some of the simultaneously movement requires a lot of dexterity. And it is worth it!