"I want the dog that is perfect for us! I want to adore it the second I lay eyes on it. I don't want to just pick up a random dog and hope that we grow to love it, hope that it suits us."
But that's exactly what happens when you get a puppy! You pick a breed with traits you think will suit you and you put in work and just hope that it ends up having those traits...but you have no idea really what that puppy is going to be like.
With a rescue dog, it's already got a developed personality, you already know whether it suits you or not.
You don't just turn up at a rescue and go - I'll take that one, a good rescue matches you with dogs that might suit your needs and then you see if you click with any of them.
I'm not saying you're right or wrong to be looking for a puppy of a certain breed, but it is much more of a gamble doing that than getting an adult dog. Also, there's no shortage of any breeds in rescues and every pedigree breed has breed specific rescues.
By all means look for a puppy, but it's no guarantee of getting what you expect.
Anyway back to your original question, small affectionate breeds - Bichon Frises, Tibetan Terriers, Lhasa Apso, Shih tzus, Whippets, Lowchens...if you're looking at Havanese there are also the Bolognese and Coton De Tulears.
Two worth a mention because they have great personalities even though there are health issues and you'd need to find really good breeders are pugs and cavalier king charles spaniels.