Cazoo are already doing this, to an extent. Their shirt sponsor is white on the home (matches the shorts), claret on the away (matches the crest), and yellow on the third (matches the crest). If you go on their website, their logo is orange.
Sponsors will never willingly opt to be on the back of a shirt without paying significantly less money. It’s always going to be much less exposure, unless players start doing post match interviews with their arse to the camera.