It is not actually a "primer" bulb, but a purge, it pulls fresh fuel through the carb, and then dumps it back into the tank, so if it is doing as you describe, you likely have a clog in your carb, or the needle may just be stuck from sitting over the winter, so
remove the carb and disassemble and clean, remove the needle/spring/needle and hold up to a light, you should be able to see
light clearly through the orifice beneath the screen, so visually verify that it is totally clear, because is may be partially clogged,
letting a little fuel through, but not enough to accelerate.
If it looks like you need a new kit or any parts, ask the Stihl dealer the price of a new carb, as many of these models, they have dropped the price of a new carb just a few dollars above the price of a new kit.