What equipment upgrades or tweaks will improve the sound of drums?

Don’t know where the OP is, but its safe to bet that, even with a sub, if those small speakers, about the size of kef ls50s, are not in a small room like what they were designed to be in, its a very bad bet that tweaks alone will do much other than put money better spent in the vendor’s pockets.

First thing first. 

No secret here.

It's not clear what your "quality vinyl recordings" are. IMO, reissues are always a disappointment if you have the opportunity to play a period release.

Reissues are fine, but only as a last resort for my ears

Seek out original presses. Your system is fine, LM gear seems to be solid, so you should be using a phono amp on on the same level for maximum vibe.

I never had a problem with drum performance when I had a P5. The plastic  subplatter, when replaced with one of the aftermarket models seemed to add more authority to percussion and bass.  

It might just not be in the original mix; so no matter what you upgrade it’s not going to happen. Maybe look into other bands or drummers such as Danny Carey of Tool or other heavy handed drummers. I don’t recall any albums of Elton John having the drums being mixed up front. 

Agree with much noted above but not sure upgrading any component or tweak is going to remaster what the engineer did in the studio. Also the theatre you saw Rocket Man in, as I did, the system is EQ so heavy on the bottom end you’re never going to duplicate that for many reasons including the size of the room and the amplification behind those speakers and subs. Not going to happen... sorry.

Seriously the best thing you should do first is work on your room if you haven’t such as room treatment and speaker placement. Everything I said above is true too. Good luck. If you haven’t discovered Tool... you should.

it was a great movie and sad how his parents treated him. I’ve always enjoyed Elton John but after seeing the movie I have much more respect for him and the struggles he conquered. Don’t forget the room treatment.

I am imagining the speaker placement isnt the best it could be . Odd furniture creating reflection . Distance the speakers the same from your side walls . Dont be against a wall in a corner or have a metal filing cabinet within your angle of reflection. I know it sounds like obvious points but its important to have the perfect sound stage before  being accurately dissatisfied. Sub woofer is going to have very little to do with good drums. Your mids and tweets are going to give the tone and life to percussion. Do i have to mention tube rolling has an immediate and significant change to drums. Be like changing out the drum kit with each tube. Soundstage perfection then  tubes . In that order. 

I think it’s all in the recording and how the recording was done .
I realize your discussion revolves around vinyl playback however you need to hear these spectacular recordings .
For 25 bucks you can judge for yourself, Tonian Labs percussion recordings are shockingly realistic though so far only available on CD .
The recording industry could definitely take lessons from Tony Minasian of Tonian Labs .