So there's some seed of a potentially valid point here. Phrased in a Bayesian fashion, if one assigns low enough priors to certain hypotheses, one isn't going to practically consider those hypotheses unless one has ridiculous levels of evidence. So is something like that happening here?

I think the conclusion is "no". There are many religious individuals who have no objection to evolution. The objections stand essentially from religions which have creation stories which are important to the theology. For example, in Christianity, the Fall is very important, and you get a lot of Christians who object to evolution. Islam and Judaism have as important theological points that there deity is the creator, but the method of creation isn't as important, and one sees less objection to evolution in those religions. Among some religions which don't have any issues of this sort, or have very weak or very seldomly directly intervening deities (such as some forms of New Age religions) one sees close to no objections to evolution (although there are some prominent exceptions such as Deepak Chopra). But in all these cases, there are people who adopt an essentially similar theological standpoint and yet accept evolution, and this includes full-out major denominations such as Roman Catholicism. Now, it is possible that they simply haven't really adapted a consistent world-view (consistency isn't a human strongpoint), but that's an argument that would need to be made in detail.

In this case you see the amazing complexity of life as a product of chance/random events and not because of some genius unseen designer. The beautiful world you are describing could be interpreted as being the product of either, and the science itself would not change.

So this actually isn't the case. Evolved life doesn't look like what you might expect from a designer. When we looked at designed objects we see all sorts of commonalities that make sense from a design standpoint: we see efficiency, modularity, and reuse of parts between designs. We don't see any of those in evolved life.

Large amounts of life-designs are highly inefficient, almost as if they were added haphazardly by evolution. The giraffe's nerves which loop all the way from the head down through the neck and then back up again to the head are a good example.

We don't see modularity- pieces don't develop and integrate separately except at a very weak level. Thus birth defects that cause one organ to not grow frequently cause problems throughout the body or at very minimum in neighboring organs.

And we don't reuse of design pieces except for the very basic aspects that form a perfect nested hierarchy. This is in contrast to for example computers. This would be akin to having only mice on one type of computer, and only joysticks on another, and the only computers with USB drives were a subset of the computers with mice that didn't overlap at all with some subset of computers with mice with CD drives. But we don't see that with computers, or cars, or any other designed object that has many different designed versions. They reuse the same ideas and technologies. This is something designers do for obvious reasons: you don't need to reinvent the wheel. But evolution (aside from some very tiny examples of horizontal evolution where a single gene or small number of genes has been copied over due to viruses and a few neato parasites like my favorite parasite, Wolbachia) doesn't have that option. Pandas would have a much easier time if they had a real thumb instead of an inefficient bone spur. A direct design would have copied over the great ape thumb over or use a very similar design.

So this really is an example where one can look at what one would expect from design and from evolution and conclude that evolution makes more sense in context. This doesn't rule out less direct intervention, say a deity setting up life on the planet, letting it evolve and then showing a few thousand years ago to talk to a desert tribe and tell them not to eat shellfish. But that's a distinct issue from evolution except in so far as that if the same text that claims this deity did intervene also claims that the deity did design everything from scratch, then that's a reason to maybe doubt the text. But this has nothing to do with belief structures or as the Bayesian would say, extreme priors. This is about evidence.