Lets start from experimental data. What is an electron, what do we know about an electron? It is too small to touch or see or smell. Everything we know about an electron comes from several levels of proxies. We end up measuring a track circle in a magnetic field and get e/m, consistently for different "electrons" and we do the millikan oil drop and get e and then we can assign a mass to these manifestations consistently.
That is all we have for the electron, it has a mass m_e measured and a charge e, measured.
Nature has been good to us and a working theory exists for QED. Mathematics is a tool, it can describe and predict measurements but it is not something that creates reality. Reality is what one measures . If the theory predicts, it does not matter if it goes into a yoga position to do so, as long as it can predict consistently. They want to call them bare and dressed mass? Fine. Who can measure anything more than that the measured mass is m_e and and the measured charge e?
Better theories/computations may come up, but to be better they should describe existing measurements and predict more and different ones, that QED cannot explain, for anybody to pay any attention. Or be as overwhelmingly economic and elegant as the heliocentric is to the geocentric pov. QED works.
added: I want to give an example from real physics history that I heard from the horse's mouth back in the 1980s, of how succesful new methods of computation overwhelm tradition and sweep over reluctances once shown to successfully predict faster and accurately.
Back in the Manhattan day project, a physicist think tank had been set up with the best brains of the time to calculate crossections needed for making the bomb. Feynman was a junior member of the team. They gave the group a problem and a week later people reported the result of their independent calculations, parallel processing. Feynman said that one afternoon he was lying on his bed with his feet on the wall, when the Feynman diagram method came to him, whole ( he had eidetic memory so he probably saw it). He calculated the current problem and waited impatiently for the report of the others. When he gained confidence that his method was as good as the long drawn out s matrix calculations he started playing games with the team. He would get the result in an evening, tell them the next day what they would find, and it would take them the rest of the week to confirm.
Of course Feynman diagrams were universally accepted after that.
I was reminded of this story when I listened to the talk of Nima Arkani-Hamed which he gave on the twistor revolution. He finds extremely cumbersome the Feynman diagrams method and is exploring a new one that gives the same results as the thousands of summed QCD feynman diagrams. I was amused, and am sure that Feynman would have been too, if he were still alive.
If a new computational method is faster, sleeker and as predictive, it will be adopted as surely as God made little cabbages.
In my experimentalist's opinion of course.