NO! Electronic drums are *not* the solution! Haha
There is a mix/room issue going on with this recording, but it's live, so not too unusual ... It sounds like they were trying to get away with just a kick mic and overheads to capture the whole kit from above, so the kit elements are kind of getting blurred together.
I think his kit has the potential to sound pretty good in a studio just the way it is. Not my preferred tuning, but not bad either.
If I was going to nitpick his sound to make it how I tune my drums (I'm a drummer 1st, bass/guitar player a distant 2nd but gaining rapidly), I'd take the mattresses out of that bass drum and tune it for a tighter sound. There's just so much more tone and control you can get from an unmuffled (or barely muffled) kick, and without getting too boomy. Think Bonham. Before anyone flames me for heresy on that, I'll point out that it was a recording engineer who turned me on to unmuffled kicks about 5 years ago and I've been doing it ever since both live and in the studio and it sounds great.
Also, his snare isn't quite tuned tight enough, especially on the snare-side, so it has that kind of soggy, deadened "thwack" sound. Tighten up the reso a bit higher than the batter and bring 'em both up about an octave to dial it in nice warm "crack" without getting harsh as overly-cranked snares tend to get.
His cymbals are simply too loud in the mix, which is a bummer cos they don't sound terrible if taken in isolation.
All this to say that tuning is primarily responsible for a drum sound. I'd say at least 90% of the way drums sound is in how the heads are tuned (assuming they're still capable of being tuned), and if, or how much, they're dampened.
It's not nearly as much about whether the drums are some cheapo early-'80s Taiwanese stencil set, or a shiny new high end maple DW kit as people (including drummers!) tend to think.
Cymbals, on the other hand ... you get what you get, and you can't tune or dial them in, which is why so many drummers obsess over them and spend thousands on vintage Zildjian Ks from the '50s.
But even if his kit were tuned to my preferences, they probably still wouldn't have sounded pleasing on that recording given the room, mic placement, and mixing. Might have sounded great in person, though.
Drums are such a PITA to record well.