Seems like you have a pretty good handle on the capacitor replacement. The "big black thing" is a resistor and looks to be well made.
I would definitely try to mount the new caps in the through holes. If you haven't had at it with solder wick, this is probably a good learning platform, lot's of free space, lead count low, and I would assume the board is amenable to the work.
If it was me, I'd check the values of all the resistors just to be on the safe side. That might mean lifting a lead from each one so may be more work than what you're willing to tackle.
I would also replace the electrolytic caps because of their age and the low price for quality replacements.

I've been using these Axon True Caps and they seem to work fine (built by Solen). I've only used them in a couple of situations but they are as good as any other film cap I've used and the price is right.

Good luck with the crossover rehab!