It does seem clear that pols for high office are not dealt with the same way as people who aspire to high security jobs in the CIA FBI and the like. And I guess that is another question I might have asked originally - should they be?
I think they should all be military veterans, but, like your question, no extra-Constitutional requirements can be instituted. If the people want to elect a multiple murderer they can, if his citizenship has been restored and he is otherwise qualified.
If there were to be security checks it would get very political in a very short amount of time. There would have to be parameters set as to what is acceptable and what is not. It doesn’t matter who does the check there is bound to be some bias and I am sure there would be a lot of things they would find on one and many they would “accidentally” overlook on others.
I think it would be a mess, but I certainly understand the point you are making. I think this particular crop of candidates likely begs your question more so than any I can recall. We certainly, hopefully could have done better than these three on our worst day. But, this is what we have and we will have to do our best to make chicken salad out of chicken $sh!t. This is the unenviable task that lies ahead of the American electorate. :)