I have a similar problem as well. I bought a pixel 10 pro XL, and changed my setup from having a blank owner profile, non google app profile, and a google profile, to a single profile with non google apps on the main, and google apps on my private space.
The first time I connected to my car with (private space) android auto, it seemed to have worked. however, every other time my car doesn't recognize my phone is connected. After reading this thread, the only solution to make it connect is by having the developer settings enabled and going into the Default USB configuration and changing it to something other than no data transfer. As soon as I change it, my vehicle instantly recognizes it and I am able to change it back to no data transfer and it continues working until I turn off my car.
Oddly enough, I noticed when its connected to my car, I don't think any setting is selected in the USB section when connected to my vehicle , once I change a setting it crashes back to the developer settings and I have to reselect Default USB configuration. Once I reopen it, no option is selected again and I can select any option and itll crash back to developer settings.
The setup I have is a wired connection to my car on a Pixel 10 Pro XL with android auto in private space, I have play services and the play store too in the private space. I do not have any google apps in my main profile. I hope we can figure something out as I rather not be having the developer settings enabled to change one setting every time i drive my car.
Love this OS, been using it for years at this point.