Physicists start considering it as nothing more than a mathematical construct to help in their equations but with little to no physical reality -- despite experiments to the contrary.

HiI recently checked out an online paper that discussed this concept. It started with a discussion of the history of Maxwell's equations and their rewriting by other physicists, plus some concepts that could be fruitfully explored for new physical insights. That was quite intriguing and is a worthwhile research topic, so well worth exploring further. The vector representation vs using quaternions and Clifford algebra is an active bit of mathematical physics - preprints pop up on the arXiv fairly regularly.However the author lost me when he claimed electron and proton beams don't have a magnetic field. By that point I knew he was a crank with no experimental experience. Magnetic self-focusing of electron beams has been studied & used for years in plasma physics. That, and other slips, indicated he had introduced contradictions into his theory - and a contradiction can be used to 'prove' anything.