A few reasons:
You usually get the whole venue, whereas with a hotel you don't. So it's a more personal event, and maybe less stressful because you aren't going to worry that your event is causing noise or inconvenience for other guests.
If you are having a whole day wedding, it's good to have the grounds for guests to get some space and for kids to run around. Hotels don't typically have large grounds (unless they are converted stately homes!)
The buildings can be beautiful and people enjoy being in a beautiful space. I'm not sure your assumption that they are pretending to be "something they are not" is valid. It's more like buying a special outfit for a special occasion - it's special exactly because it's not your everyday.
I wonder if you are assuming "stately homes" is something people look for, when what people actually look for is "wedding venues" , get a list of places that do weddings in their area, and decide which one they like best.
Mostly likely many of them are stately homes because the UK has a lot of them and they make good venues for all those reasons about, but the list might also include breweries, museums, hotels, barns, former schools, golf courses, sailing clubs and many others I haven't thought of. So it's not so much that people are deciding "I want my wedding in a stately home" as they are looking for a venue and finding a stately home is the best fit.