and here's a little insiders trick.
middle of the bottom of the pump there is a threaded hole. I think its 1/4". screw a bolt in that hole. find a piece of thin wood (preferably hard wood), and lay it on the edge of the hole. And then use your pliers to pry up on that bolt.
Once seal is popped loose, remove wood and pliers.... should be able to pull out by hand.
I'm hoping your diagnosis is correct!!
In the years since I did my first one, I've inherited from somewhere a very simple DIY tool that works really well, if anyone is interested.