Unfortunately, the game supports only 2 sets of icons ; the PS4 controller is still ok enough with the PS5 icon set, though.
You could select the default Tag if it's clearer for you. (it's only a visual help)
So from your Bug Report, when Steam Input is on, there are 0 controllers detected, so I guess you had the "Controller to keyboard" Steam Input profile selected (like you said). Why it'd stop to work is a mystery to me, though...

If you want it back, try to click re-activate Steam Input, then click on Steam Lib -> TE4 -> Controller Layout (it may need a refresh to appear) and then deselect & reselect the correct layout.
With Steam Input off, the game detects & seems to handle correctly your PS3 controller. I'd recommend keeping it, because you can enjoy the analog direction for running.