Standard QED correctly encodes all finite time physics about electromagnetic processes in its renormalized equations, consistent with relativistic and gauge invariance; and almost all in the asymptotic domain, where the infrared problems are partially but not fully resolved.
So any proposal to reform QED must be measured against these achievements.
I am not blaming that your models are not relativistic and gauge invariance. But I am blaming that you seriously consider your toy models to be a big step towards a novel QED though neither you nor anyone else has the slightest idea how to make the step from the toy to the real thing. The path you propose is covered with unsurmountable barriers called relativistic and gauge invariance. But you don't care and happily repeat your proposals.
If QED were a mountain top in the sea of relativistic and gauge invariance, your toy model is like a small rocky hill on the other side of the shore, that you discovered how to climb. And, knowing that the famous climber Dirac thought that the current difficult path to the top of QED could not be the final answer, you shout, ''Hey, I finally found what Dirac was missing - the principle by means of which one can climb the QED top'', without having the slightest idea of how to cross the sea. Why should anyone take you seriously?
One must begin at the right side of the shore - assuming relativistic and gauge invariance -, and then see how to find from there a better path to the top. But this cannot be done without getting from the literature the best charts that already exist, and studying them in more detail than the others, so that one can discover the uncharted territory there that might hide a better way to the top. And yes, if one then has better equipment than the Elders, one might be more successful.
So if you say,
You cannot imagine how many relativistic- and gauge-invariant expressions can be written down, but the main difficulty is in correctly encoding physics in the new equations,
I fully agree, this is indeed the essential problem. But instead of suggesting a new way how to approach this problem (which would count as a step towards a novel QED) you throw away the first, essential half - you don't have QED unless you have these! You replace it by another problem where the essential difficulty is absent, solve the resulting easy problem, but claim you are close to a novel QED.
The essential difficulty is not to cope with short-distance physics or with the photon dressing. Here your toy models indeed propose techniques for handling these. Not the only known techniques by the way; for example, you might want to look at the work by Derezinski, who spent most of his academic career with various aspects of such problems.
The essential difficulty is rather to find a way to formulate a relativistic and gauge-invariant theory in which the short- and long-distance problems can be handled better than with the current, successful but somewhat awkward renormalization techniques. In approximations without relativity or gauge invariance, the problem has been already solved by people like Derezinski, with models far closer to real QED than your simple toy models.
Thus there is no shortage of methods for handling short- and long-distance problems, but only of methods for combining these with relativistic and gauge invariance. You contributed nothing to the latter, hence nothing towards a novel QED.