Welcome to YT
It's true that there may be rescues who are there because of 'issues', but I think there are many more who are being rehomed for reasons such as the recession, or death of the original owner etc. When I was looking to get a yorkie I saw some adorable rescue yorkies and yorkie mixes. Unfortunately I wasn't eligible to rescue because I have young children and it was against the shelter's policy, but I certainly would have taken that route if I had been able to.
Good luck to you and your future furry family member
xxx