I think I understand your problem. When a tooth is extracted, the architecture of the gums and bone around the adjacent teeth changes during healing. This is called a "periodontal defect" when it heals in a way that the gums have a deep, uncleanable pocket. A 2-3 mm pocket is cleanable with a toothbrush, but the bristles won't reliably go much deeper (some say up to 5mm depending on location). This is very common in people who wait to have their wisdom teeth (3rd molars) extracted after the age of 25, because the bone does not heal back as well as the gums beyong age 25, which allows a pocket to form between the gums and the tooth. However, it can happen at any age if good hygiene isn't maintained in that area. This is one of the most common areas because the back side of a back tooth is very difficult to clean since your toothbrush doesn't fit well into that area.
If you have 6-7 mm deep pockets, there is no way you can clean them. The "infection" is real and is a product of oral bacteria living deep in the pocket without being cleaned. If left untreated, this pocket tends to spread around to the sides and front of the tooth which severely lessens the prognosis of the tooth. Eventually the infection will show up on other teeth if left untreated. The bleeding is inflammation of the gums which is how the gums respond to infection. These pockets also tend to hurt more than healthier areas when she probes them with her measuring stick. You may think, "of course I'm gonna bleed if she's sticking that thing down in my gums" but you only have to touch inside the pocket to get the bleeding. That's how we distinguish healthy from infected gums.
The first treatment is always to have the deep pockets cleaned professionally with instruments that can reach the bottom of the pockets. This is tried first because many pockets will respond to this and heal once the bacterial load is minimized. My guess is that's what happened in your cleaning 6 months ago at your first appointment with this new dentist. Now they can see that the pockets are still there, so it's time for the next step which often includes surgery to eliminate the pocket. The one procedure that comes to mind is a "distal wedge" but some of the Perio people here could probably give better info on that stuff.
I've never heard of Artisan and I couldn't find anything when I google'd it. My guess is that it's an antibiotic which is placed down in the pocket. I think I remember people using tetracycline wafers when I was in dental school. It may be a good treatment, but the dentist may also have a big box that he bought and is trying to get rid of it. Before you do anything, you should understand that many times this problem will still need a small surgical procedure in the future if the pocket still remains. But this is much better than the alternative (extraction) if you don't do anything.
As for the money, $79 is a great price for ANY dental procedure these days. Don't get to hung up on insurance because dental insurance, as a rule, sucks.