Re: I was just offered a 1977 stratocaster
The other guitarist from my band of the early Eighties owned (and still owns) a 1976 hard tail Stratocaster. It was heavier than it needed to be but played nicely. Thanks to a set of Velvet Hammer pickups, it also sounded great. My friend was lucky, the neck on his example was a good fit.
Looking at the upper photograph in the opening post, there is a conspicuous black area between the body and the Low E side of the neck. This hints at either poor neck alignment or indifferent neck/pocket fit. (Annoying but correctable.)
Since this particular Stratocaster is an exchange offer against your Jag rather than a cash purchase, there is an argument for accepting the guitar then putting some work and money into it. Worse case scenario, you can be confident of selling it on.