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Yes, the metal contacts in those plastic bodied connectors DO come out. {They had to put them in there, didn't they? } Usually, the contact pin is retained in the plastic piece with a metal tab that snaps in place when the contact is inserted. You can use a small [jeweler's] screwdriver to press into the 'wire' side so that it can push that metal tab down flat. Then you can push the wire and contact out the end of the connector.
Call around to some of the local car parts stores. Some will carry the contacts and various connectors. Once you see how it is assembled, you can repair it yourself very easily.
I had this happen on my '77 Trans Am in the '80s. I clipped off the plastic connector, got two female disconnects, crimped those on and made sure they were on the correct terminals.
That connector (like many of that era) uses the Type 56 female terminal, which are sold at Pep-Boys or NAPA. As said, release the tab on the terminal, push the old one out of the housing, strip the end of the wire, crimp (and I always solder as well) the terminal on the end of the wire, and push it into the housing to engage that locking-tab. Cheaper, easier, and cleaner than the pigtail route.