I think the cone is just going to die, like you say, sooner rather than later as it is going to be used professionally.
Hard to evaluate without seeing it.
I've got a JBL D-130 (not "F") with an original cone that was probably made in late-50's early 60's. I have zero reservations using it with a pair of 807's.
The D-130F was a D-130 rethought for the musical instrument market, with the biggest difference being extra "goop" on the speaker surround to help it hold up in outdoor use. These were intended to be battered by the Vibrasonic and the Showman amps (and popped up in other Fenders too).
The Jensen speakers used in Fender were, by comparison, junk. Fender used Jensen's cheaper speakers in amps, and the JBL was always more expensive and much better made than the Jensen line.
I wouldn't worry about blowing the voice coil of this speaker, unless you see something that makes you think the speaker is compromised in general.
Would it be worthwhile getting the speaker remagnetised?
I don't see why. They'e permanent alnico magnets, and again... unless you're seeing something implying obvious damage or degradation, I'd leave it alone.
If you're used to Jensen speakers, and 12's in general, this speaker should surprise you with deep low end and quite a bit more treble extension that you're used to hearing; more so if it has the aluminum dust cover.