Same here. No solutions so far.

Kao 2 is not from 2003, but highly updated Steam version (in GOG\ZOOM DRM-free form). Still not perfectly support xinput. Kao1 and Kao3 is old dinput games. Dinput mean you ALAYS shouldmanually configure controls, no hot-lug usually, support for analog stick is not guaranteed - most of features is game based, while native xinput mean all features is taken from DirectX + default controls scheme if natively for Xbox pads. No defaults scheme for ant else dinput games\pads.

Games from that era on PC had a square box for the stick, not round like modern ones.
Both axes can be moved to their full extend at the same time, not just in a circle, which means that the full extend in a corner is 1.42 (sqrt(1+1)) and not 1 like on modern gamepads.
That's because other than on other systems flightsticks were quite popular on PC, allowing full movement for pirch and roll at the same time.
https://upload.wikimedia.org/wikipedia/commons/c/c1/Yaw_Axis_Corrected.svg
USB versions of these analog sticks and also the first gamepads would allow the same movement.
So if a game expects 80% extend for fast movement, the 71% provided by modern controllers won't do.

For games like this I have a old Logitech Rumblepad II, bought a second one on Ebay, just to make sure.
https://m.media-amazon.com/images/I/41WBS12EMBL.jpg

There are some gamepads that try to compensate this problem however, I accidently stumbled across one looking for a replacement of my XBox360 Gamepad
https://www.amazon.com/gp/product/B07G13KQ57/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
This one also has round movement, but the axes go beyond how far you can actually move the stick.
So the movement area looks like a circle with flattened sides or a square with flat rounded corners. It works surprisingly well for older games.

In my opinion games should have an option to select which type of movement is wanted. They had to have something like this internally, since on XBox they had a round area an on PC a square one.



ps: The game where I first noticed this was Ultimate Spider-man. Also the first original Lego Star Wars game behaves like this.