Either way works for replacing caps. I've done both ways, and it just depends on how microscopic I want to get. The perceived results are always better, the more effort involved :-) The actual results, assuming tight connections, of course, are pretty much equivalent.
I don't see any harm in opening up some space around the crossover. I'd be worried, actually, on finding melted *anything* inside a speaker. It's not that common, but it does happen, often capacitor-related.
You have the early style Six, which was manufactured until sometime in 1963. The drivers are epoxied in. Replaceable drivers came after that. One way to tell the difference is that the later models had a recessed connection/crossover plate. The later style was sold until sometime in 1972.