Can my local primary care doctor treat baker's cyst?

Orthopaedic surgeon. Most baker's cysts do not require treatment. They are usually a secondary sign of some injury inside the knee joint, perhaps arthritis or a meniscus tear. Unless the cyst is so large as to put significant pressure on the nerves and blood vessels behind the knee the cyst rarely needs drainage or excision. Treatment should be directed to the underlying cause.
Yes. The "baker's cyst" is just a balloon type of pocket of fluid that is connected to the knee joint. It is a reflection of swelling in the knee caused by arthritis or meniscus (knee cushions) tears. When the knee becomes irritated, the fluid can build up in the "baker's cyst". We don't routinely drain or take out the cyst; they generally return. Mostly, we leave them alone and understand them.