Try disabling cookies. This exactly describes the effect, except that it's not mysterious once you're used to it. If something doesn't work, I open it in Chrome (where I have cookies enabled), and it works.
In many cases, I suppose the developer doesn't know about the cookie dependency (because of a framework or some other dependency). In other cases, I guess they don't care. Rarely does the page actually tell you that cookies are required.
That is a terribly user-unfriendly design decision because it does not distinguish between failure modes. Maybe your site is broken because I am running noscript or maybe it broken because it is mis-configured or recently hacked or my ISP's proxy is black-holing things or a dozen other things that could go wrong.
The point is that it is relatively trivial for the developer to add automated checks for per-requisites and display warnings for the ones that are missing. It is a lot harder for each user to manually run down through the list of all the things that could go wrong.
Just don't make the warnings into roadblocks. Inform the user and let them decide to proceed or not.
Actually, it's possible to implement in it an unobtrusive way that makes users feel more comfortable. The huge popups usually comes from the same websites that do popover ads (that commonly end up invisible and blocking scrolling with adblock) and auto-play videos.
Most of the time it's a small bar on the top or bottom of the screen. Stop whining.
Again, blame the web designers who don't even do enough diligence how others do the same. And demands of proper, obvious labeling is one of the better tools to stop companies from putting useless crap everywhere.
I don't see this. Web is filled with advertising, new laptops are filled with crapware. Labeling didn't do nothing to stop these things. What we get from lawers on the web are idiotic terms and conditions, "I agree" buttons under the wall of text, copyright craziness with idiotic cease and desist, persecution of people publishing secrets or just snooping around. I think if lawyers kept there noses out of the web it would be much safer place.