which GPS devices accept GPX files?

All modern automotive devices accept .gpx files, but only certain ones can be read. Your devices can read routes and waypoints from BaseCamp, but not tracks for example.  What can't you find on your devices?

It seems my devices might "accept" (aka import) a GPX file, but I can't seem do do anything with it on certain models.

I don't understand your comment "Your devices can read routes and waypoints from BaseCamp, but not tracks".  Can you please elaborate?

How can one tell which models can actually follow a route from a GPX file?

Tracks in BaseCamp are different to routes. Two different tools to create them, you need to be using the route tool in BaseCamp.  If you send a route to your device it will appear in your trip planner app.

I seem to be asking the wrong questions so let me state it another way.

I go into a store (or online) to purchase a Garmin automotive GPS.  How can I determine which model will take the GPX file I created in Basemap, and navigate my drive along the pre-determined route?

My "modern" RV-BBT600 didn't want any routes from BaseCamp, which was also confirmed by Garmin support. Now I'm going to take my Navigator V again. But now I'm looking for a device that interacts with BaseCamp and where I can enter the height, width and weight of my vehicle. Which one can do it?

Yes unfortunately Garmin has now introduced models that don't work with BaseCamp. Note however that while older models take routes from BaseCamp entering things like vehicle height etc only works on the device. Which devices that do that though isn't really a BaseCamp question and is best answered by looking at the current model line up.