What's the alternative?

As I see it, the possible ways to think about it are as follows:

1) You must think of it as 50/50 possibility of being true until an absolute proof is found.

2) You look at places where it could fail, and if it passes then you update the likelihood based on that.

To me (and as this post argues) it makes sense to follow (2). Not only is it more useful practically (you can use that assumption to solve other things, conditional on it being true) but also it reflects the reality of the world. Even if it turns out not to be true in every case it is clear that in the majority of cases it is true.

Maybe it is untrue but it is possible to define the set of cases where it is untrue. If this is the case, it seems likely that class is very small, so it is mostly true. That is the opposite of a proof of course, but it does seem reasonable to think about your confidence of it being true as being somewhat proportional to the likely size of the class.

close complex border that seems to yield increasing numbers of "close calls" sounds exactly like the situation where there's a lurking, sparse class of "touches" that we haven't thought of

That's not really how it works though. If it keeps happening and those classes don't touch, then it typically means they aren't as close as they seem (ie, there is an extra dimension in which they are a long way apart). In this case it seems like the P/NP divide is actually a thing which means things which appear close actually aren't, so if you analyze them in that space they don't appear close together at all.