There was a short point in time when the doors at keeps/outposts in Cyrodiil were working properly.

ilander

So, ZOS did a fix at some point in time in the relatively distant past because so many people had been complaining about the doors not working in Cyro when you pressed the Interact button.

I'm curious to know why this only lasted a couple of patches at best because now you can't get in a door at all a lot of the time despite pressing the interact button tens of times.. and, yes i do mean tens of times.. I think I've possibly done it as much as 40 times before without it working in the last few weeks and i know for a fact that this happens to everyone so it is not my internet (or my router) nor the spec on my pc that is causing it.

Also before anyone says it.. no, it is not a desync and i'm not already dead.

You can clearly see the interact E depressing on your screen repeatedly so where exactly is this process breaking down that it won't actually work the door when it clearly did work first time for a good while after their fix back at the end of 2015 or so (don't remember the exact timescale for it but it is more than a year ago now.)