UPDATE
Pulled the engine cowling and chatted with an A&P. We suspect the leaking vacuum pump cover plate might be the culprit because it leaks oil directly onto the exhaust muffler. The burning smell in the cockpit did not register as smelling like burning oil to either myself, a CFI, nor a 20,000 hour pilot I went up with, but perhaps that's what it is. We found a bunch of oil on the bottom inside of the lower cowling when we removed it. My best guess at this point is, when the airplane rotates on takeoff, that causes the oil to then drip more aggressively right onto the exhaust (you can see in the photo that there are drips hanging from the screws and I'm imagining on rotation, that drips onto the exhaust). I actually just had this gasket replaced about a month ago because it was leaking. And now it's leaking again. And upon checking the engine logbook, this gasket was also replaced 8 years ago, just a year or two after the plane was brand new. So for some reason this damn cover plate doesn't want to seal. I almost wonder if there's any reason not to permanently seal it with some kind of high-temp sealant since the gasket does not seem to be doing the job for whatever reason.