It's First Direct
https://www1.firstdirect.com/It's a Telephone Bank in the UK, launched in 1989 and I became a customer a year or three after that. Because it doesn't have any branches its call centre staff have to be trained to handle absolutely anything - if they can't fix it then it won't get fixed.
I found it because my father used it, I have no idea why, he was not ordinarily a man to favour technologically sophisticated solutions, he never owned his own email address for example. Maybe as a working man he found it frustrating that other banks were closed after hours? First Direct is never closed, it operates 24/7. I have used other banks for some things, but I've always kept accounts with First Direct because of their truly extraordinary customer service hence I call it the "good bank". I actually know one of their Founders and apparently that commitment to customer service was key to their original vision for the bank, he led a strategy session for the start-up where I work now and it used that vision to give us a worked example. He was Chairman at another start-up I've worked for too. Small world.
For the buying a home part I do mean that I bought it outright by the way, there wasn't a mortgage or anything like that - so that's a lot of money, let's say six figures. I presume the precaution was triggered by the fact that I wanted to move this large sum (almost all the money I had) to an account I'd never sent any money to before. Sounds almost exactly like a scam.
I would _like_ to believe any other bank would have similar protections - but of course I don't buy a home every day, so I have no other examples to compare, and most of my contemporaries have a mortgage so the very large sums aren't involved.
The password when they call me thing was nice, to be honest they didn't proactively set that up. I suggested it off-handedly one day and they were like "Of course, yes, we can set that up". I presume it's not custom, just they didn't explicitly advertise it to me as an option. That happens a lot, I didn't want contactless on my new card recently, and they were like "Yup, of course, done. Replacements for this card will also not have contactless until you call to change that".