I started out on the path of rebuilding the carb so (hopefully) it wouldn't leak fuel when it wasn't running, but in the end, based on comments here, and the fact that to get the e-clip on the throttle out with what I had, I ended up having to remove the top shield and thus two head bolts, I ended up going more or less all out and pulling the head. Here's what it looked like (Click the images if you want to go all out full resolution):
These are as the head was removed. Nothing touched otherwise. I can't claim to be an expert, but it appears to me as if the HG was blown towards the front of the tractor.
I've since cleaned it up - scraped the carbon off the block (it's hard to tell in the pic, but the carbon buildup on the block was pretty significant) and cleaned up the head. I'm reserving final prep of the block until I have the new HG to put it back together.
I've watched a Youtube video on flattening out a Kohler head, which I used the info from on cleaning up a B&S head for a pressure washer that I had to pull the head on, but the glass I used for that broke by the time I was done. It just dawned on me that I have an 18"x18" absolute black granite tile doing nothing that appears to be really flat. And more robust than a 1/8" thick sheet of class from Home Depot. Think that might be adequate to do the same with the Kohler head?
I'd like to do a little more on the head, if possible, since brushes can only get so far - Is there any reason I couldn't stick it in my blast cabinet and clean it up more that way (I use baking soda in my cabinet)?
I still strongly suspect it needs more than just a HG, and to that end I'm debating taking this motor to J-Mech and having him rebuild it, or bringing another K301 that's still at my grandmothers back, trying my hand at a rebuild of that, and if I should fail at rebuilding it, then take this one to him.
That all being said, any other recommendations form the pros here? For the short term, I'm going to put it back together to help deal with leaves between now and when 'Fall' becomes 'Fell' (I still need to figure out why the deck doesn't want to hang level, but that's a topic for another thread). But after that, I'd like to try to pursue further improving/restoring it.