I watched PBS Spacetime's video on virtual particles (link goes to relevant timestamp) where they say that virtual particles shouldn't be thought of as anything more than math because they aren't even mathematically necessary, as the lattice version of nonperturbative QFT doesn't use them, but still makes all the same predictions as perturbative QFT. I was satisfied with that, until I had a brief exchange with someone in the comments of this answer where he says that, in most cases, it's impossible to actually do computations in nonperturbative QFT, and, when I asked if this was just due to not having suffiently efficient algorithms, he said

Note that in particular that even establishing the existence of a non-perturbative Yang-Mills QFT (which is what the standard model / QCD / QFD are) is a millennium problem.

which implies that we don't currently even have a nonperturbative version of the standard model and that it's unclear whether one exists.

So does that mean that, at least in the current version of QFT, virtual particles are a mathematically necessary part of the theory, not just a computational tool? But if that's the case, what do PBS Spacetime and various other sources mean when they say virtual particles are just math and shouldn't be thought of as something physical? And in particular, as PBS Spacetime is typically a reliable source, I wouldn't have expected them to mention nonperturbative QFT as the reason not to think virtual particles are physical if that theory wasn't actually useful for nontrivial calculations. Is it that most physicists think there is a nonperturbative version of the standard model and it just hasn't been discovered/created yet?

Note that I'm not asking about whether virtual particles are "real" or not in some philosophical sense. I just want to know if our current best models require them in order to make accurate predictions.

For context, I watched PBS Spacetime's video on virtual particles (link goes to relevant timestamp) where they say that virtual particles aren't mathematically necessary, because the lattice version of nonperturbative QFT doesn't use them, and yet still makes all the same predictions as perturbative QFT. I was satisfied with that, until I had a brief exchange with someone in the comments of this answer where he says that, in most cases, it's impossible to actually do computations in nonperturbative QFT, and, when I asked if this was just due to not having sufficiently efficient algorithms, he said

Note that in particular that even establishing the existence of a non-perturbative Yang-Mills QFT (which is what the standard model / QCD / QFD are) is a millennium problem.

which implies that we don't currently even have a nonperturbative version of the standard model and that it's unclear whether one exists. However, PBS Spacetime is, in my experience, typically a reliable source for high-level explanations, so I wouldn't have expected them to mention nonperturbative QFT as the reason not to think virtual particles are physical if that theory wasn't actually useful for nontrivial calculation. Is it that most physicists think there probably is a nonperturbative version of the standard model and it just hasn't been discovered/created yet?