If you bug physicists enough, they will admit that the standard model has some problems, like the Landau pole. However, there are toy QFTs in 2 spacial dimension that have models rigorous enough for mathematicians. That should be adequate for philosophical purposes.
I don't think the Landau pole can be characterized as an actual problem. It was considered a problem for strong interactions, but we now know that quantum chronodynamics is asymptotically free, so it does not have a Landau pole. The Landau pole for quantum electrodynamics is at an energy scale much much higher than the Planck energy. We already know that we need new physics at the Planck scale, so the lack of asymptotic freedom in the Standard Model is not a real practical (or even conceptual) problem.
From the last thread:
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