It seems that you haven’t tried to calibrate the controller in windows? Those “unnamed” controllers had plenty of issues to set them up at least that was the case some years ago. I struggled with a few logitech ones. The only way possible was to just install their drivers and then try to emulate xInput with various stuff.
You tried JoyToKey, but I have no idea if it was made better during the years, but it was pretty meh back in the days and taking keyboard inputs and slapping them on the analog sticks never led to good results.
I see you are on windows 10 so it will be an easy search for “Set up USB game controller…” in the windows start menu and when it’s plugged it will be in the menu. I hope there it will say mark it as Controller (XBOX 360 for Windows) with Status “OK”. That’s if the drivers come with xInput emulator build in but I doubt that.
Click Properties and see if everything is working in the TEST tab (it will default to it anyway). The right Analog stick should be X Rotation bar for horizontal movement (increasing it or decreasing it right and left) and Y Rotation bar for UP and down. The triggers should be the Z axis.
If something doesn’t work go to Settings Tab and Calibrate. If you see flickering of some things in the Test view … it’s usually ok because games will compensate with “dead zones”, but calibrating should fix those as well. Follow the instructions there and tell me how it goes.
P.S. I’m using Dual Shock 4 v2.0 for PS4 with DS4Windows.exe which simulates xInput controller and I never had any issues with it. The steam controller settings don’t include fancy features at all so I’m better of seeing buttons marked as xBox controller than sacrificing useful things.