Last dog I had was from a rescue centre. Despite the centre staff pushing Whippets & many different forms of Terrier, and after 4 visits, my dog, a border collie eventually chose me. She was lying in the corner of the kennel, pretty much ignoring everyone, including staff. I'd walked the length of the kennels to see what was available & passed her once. The staff said she was pretty friendly when she got to know you but was wary of strangers. On the way back to the reception, she stood up & came towards me. After some persuasion, the staff agreed to let me walk her, supervised) into their exercise field. When let off the lead she barely left my side.
She stayed that way when I was around for the 10 following years I had her, barely leaving my side when we were out.
Would have another dog but my elderly parents have an Airedale so should anything happen to them, he comes to me. As he doesn't get on well with other dogs (he has ADHD), it wouldn't be practical for me t take on another rescue at the moment.