The problem with the permit vs implies is if it only permits then certain parts of a theory are not testable. The solution explicitly requires those things to exist in order to fit into the rest of the physics models we have.
Sorry, no. A theory is testable if there is a way to show it's wrong. A theory may be consistent with a particular thing existing but that doesn't mean a lack of evidence for such a thing disproves the theory. I mentioned tachyons as an example.
And you mentioned white holes. They are a counter-example to your claim that "[a] solution explicitly requires those things to exist in order to fit into the rest of the physics models we have." Not only do they not need to exist in order to validate Einstein's theory, their existence would violate the laws of thermodynamics. They are considered to be a non-physical solution to Einstein's equations.
are [tachyons] permitted or are they explicitly required to exist to make the rest of the model work?
Permitted, not required. Nobody has observed tachyons, and yet special relativity has overwhelming support from experimental evidence.
Implying the existence of something I think is a better phrase
Okay, I'll buy that. But you go on...
since in many cases if you remove that element from our universe then the model ceases to make sense.
Well ... mostly no. It depends on how crucial that "element" is to the theory. Tachyons are permitted under special relativity but have not been observed, and yet Einstein's theory of special relativity survives just fine without them.
Perhaps an example of what you mean would be the luminiferous aether, proposed to explain how light can propagate in a vacuum. The Michelson-Morley experiment put the dagger in that model. The aether didn't exist, therefore the aether theory was wrong.
Now maybe there's another answer for why we don't see something, e.g. it existed at some point in time (maybe nano seconds after the big bang) and disappeared, but to say something is permitted is the wrong approach. To validate a model we go searching for that thing.
Or look for forensic evidence that it existed in the past. And saying something is "permitted" is the first step towards looking for it, so it is not the "wrong" approach. You may or may not find it, and that may or may not invalidate the theory, depending on how important that element of the theory is.
There are cases where a particle was predicted long before it was ever observed, because it appear to be "required" rather than just "permitted." The neutrino, for example: it was needed to conserve angular momentum in nuclear reactions. Neutrinos were extremely difficult to detect, hence the 26-year gap between Wolfgang Pauli's proposal in 1930 and their detection in 1956. Arguably, there would have been a serious problem in nuclear physics if it had never been found.
You misunderstood the point I was making in my post. I'm all for those models not making sense (to me), god knows many have been proven correct even in the face of extreme criticism even among physicists at the time they were proposed. I'm just pointing out to the OP that something not "feeling right" is pretty much par for the course in physical models of the universe, both among lay people and among physicists themselves.
A theory or model "not feeling right" is often the stimulus to look for better ones. But usually that stimulus comes from evidence that a theory is wrong, either in a specific situation or in general.
