Advantages: Any cat you take in, you have saved their life. You may find a total gem of a cat. And yes, there are plenty of purebred Persians in shelters, from 3 months to 15 years. Most cats are about $50, with huge discounts for older animals.
Disadvantages: The cat may be sick -- even terminally ill because the former owner could not face putting the cat to sleep. I once drove to see a white cat, but when I got there the shelter vet tech told me he was suffering from advanced kidney failure, dementia, and incontinence. It was sad to leave him, but he had no future and, the only reason I left him, he seemed unaware of anyone or anything around him. Sometimes the illness is apparent, but sometimes it is not. Chances are almost 100% that the cat will get a respiratory infection. You will know nothing about the cat's history. And something that is bad in my world, the cat will be vaccinated whether you want it to be or not. Your choice of cats is limited to what is available at the time.
Advantages: The cat will already be screened for major health problems. Some cats will come with histories and/or papers even. Most cats come with a guarantee for health and a return policy if the new cat doesn't work out. Since rescues can remove the best cats from shelters, these cats are in better condition, younger, and friendlier than from public shelters.
Disadvantages: The price can be the same as for a public shelter or considerably more. Some private rescues have adoption fees of $300+ for a purebred. Some of the information may be wrong. Spencer was adopted from a shelter -- the owners had surrendered him and said he was 2 years old, but that they had him for 3 years. Like with rescue, many of these cats will come with an URI.
3. From a Breeder
Advantages: The person will know exactly where the cat came from. If it's a kitten, then the person will not have to deal with problems created by someone else. The cat should be healthy. The breeder is usually a great source of advice on how to care for the cat/kitten. If something happens and the cat needs to be rehomed, most breeders will take their cats back.
Disadvantages: The cost is higher -- $300 to $1000 for a pet quality, more for a show quality. (Retired queens are usually rehomed for much less.)
As you can see, I have cats from all three sources. I don't see one being better than another, although I would caution a new-to-cats person not to knowingly adopt a sick cat from a public shelter. (It takes years of experience to be able to nurse some of these cats back to health.) Each has it's pros and cons.
And I'm not one to say "You must rescue cats -- not buy them." My only requirement is that the cat is loved and cared for the remainder of its life. Too many people adopt or even buy a cat, forgetting that getting a cat can be a 15 or more year commitment.