Weird WordPress browser glitch, looking for help to ID the cause

Thanks in advance to anyone who might be able to help ID a weird browser glitch.

About a week ago, I noticed that one of the less-common browsers I switch through to test site settings was showing the site's header and main page in mobile mode. (This web build, inherited from others, is 12 years old and filled with custom pages, so that inconsistency isn't so surprising.)

Then another user flagged the same problem on a different browser. For them, on Edge, zooming could knock the site's main page back into desktop mode. Then another, on Chrome, had the same issue, and as with me on Opera, zooming didn't help. Nor did trying to force a user-agent change in Dev Tools, or clearing the cache, or reloading our browsers.

Today, my Chrome is picking up the same issue (it didn't before, and my Edge is still fine) which has me thinking there's a backend change that's going to keep showing up everywhere as local caches update.

But I've checked the usual spots in our CSS, looking for browser override issues and varying max/mins, along reviewing related php/js/less files for other surprises, just to see if anything can force the browsers to read the site in desktop mode on the desktop again. Nothing changes the behaviour.

What's really frustrating is that I am just about to start a two-week window dedicated to a full rebuild of this darned site, so I don't want to waste time on a weird glitch on the old build. But if this behaviour is spreading, we're going to have more clients with issues in the coming days, so I have to find a fix now.

Where should I be looking for a solve for a problem of this type, especially since it doesn't seem to be something most users can fix on their own? Any idea what's going on here?