... what difficulties physicists have had constructing 4d axiomatic qfts?
I have my own take on this subject. Physicists are stubbornly trying to axiomatize (dogmatize) wrong guesses of interaction. R. Feynman and P. Dirac insisted on better guessing rather than on dogmatizing wrong guesses. So did J. Schwinger. Modern RG things are nothing but tentatives to freeze wrong guesses and have the same ill defined QFT. As famously said J. Schwinger:
"This way of putting the matter can hardly fail to raise the question whether we have to proceed in this tortuous manner of introducing physically extraneous hypotheses, only to delete these at the end in order to get physically meaningful results. Clearly there would be a great improvement, both conceptually and computationally, if we could identify and remove the speculative hypotheses that are implicit in the unrenormalized equations, thereby working much more at the phenomenological level. ...
I continue to hope that it has great appeal to the true physicist (Where are you?)."