Hey Kevin, my advice would be the following:
1. Spend just 79$ into decent drumheads (see for example
this drums heads pack). The one showed on the pictures are no doubt over-dead. It's like, you were trying to play guitar with strings played for over 20 years: would you?
There's nothing wrong with using pinstripe heads, but they need to be new (or they sound more precisely at their best after they've been played for like one week).
If the batter head is well tuned, a good sound engineer can get a hell of a sound out this old Slingerlands. These drums were, and are still amongst the best ever built, they can reach astronomical prices on ebay. And he seems to have nice cymbals (Zildjian A) and hardware (DW), so on this side there's not much to say.
2. Try to convince him to put resonant heads on his toms.
3. If he doesn't want to, well there's also nothing wrong with not using bottom heads! As other guys said here, it was even very popular back in time. Since you can insert the mics inside the toms, they capture the sound differently than from what is most commonly done today (a mic over the batter head plus ambiance microphones). Inside the tom, the mic gets the punch of the batter head plus some of its resonance, that you wouldn't get so distinctly outside the tom. It's a different sound (of course without all the sustain you would get when having a resonant head), old school, but in now way worst that's just a matter of taste. And I think it can suit very well to your style of music.
4. If this guy uses no bottom heads, he should then remove the bottom hoops, and even the bottom lugs out of his toms as it can generate some parasite noises.
5. Tune his snare. Generally, just applying some higher tension on the snare head helps. You can maybe borrow a drumdial to someone, it helps a lot.
6. The playing: He has some kind of feeling, but he's not in time!!! Try to make him play in the tempo, with the others. Also I noticed that he's inconsistent in the dynamics (heard some suddenly hard beaten crash of hi-hats when they shouldn't). By the way, very nice music.
7. Make a CD out this recording session, find gigs, and find eventually a new drummer.