I don't have any clips handy. For sure the tele sounds like single coils, though most people would call the tone "glassy" or "twangy" rather than "thin". It is what it is. You can dial the tone controls for more low-end and less bite. Personally I like the bite. My humbucker guitars sound dull by comparison. Teles are definitely twangy.
But it's true that the chugging is not as solid with the single coils. Humbuckers definitely help there. You could also try a BB Preamp set for a bit of bass boost and/or a bit of treble cut. That fattens up the sound a lot, and I use a BB Pre a lot, especially for leads to get a Mark tone. I also use an OD808, but that actually adds more bite. Between the two I get a lot of variation, and can keep the recto gain a bit lower for good definition. Too much gain = less punch and clarity.
One thing that Mark players often do is cut the preamp bass and bump up the low frequency post-gain with the EQ. You might try to emulate that with an EQ in the loop. The single coils should be able to maintain clarity and bite and stay tight with a lot of gain, and then you can dial up more thump with the EQ.
You can also look at alternative pickups. If you have '50s style low-output pickups that are designed for a lot of quack and clarity, you might swap in a higher output, twangier pickup for more punch.
I definitely use the recto a lot more than the Studio Pre. I prefer the grainy, punchy recto tone to the smooth Mark tone, and get close enough to Mark tone when I want it with the BBP. I do love the SP, though, when I want smooth-sustain-for-days...