Briggs 31P777 in that tractor - may need valve adj. ....Perhaps Walt or Mownie will help out - I don't know details of that procedure.

Yes, the valves out of adjustment can cause the problem.
And, if you are wondering about my addition to the heading, it is to avoid getting S-canned for answering under the original question!

So maybe there is an abnormal amount of compression due to valves out of adjustment, making it difficult for the starter to crank it?

That is correct, unless the valve adjustment is precisely where it should be, the compression release action does not work well enough for the puny starter to push past the compression stroke.
e-mail me for my version of the correct way to set the engine to the correct place for the valve adjustment.

Chrissy: Also Extreme Carbon build up within the combustion Chamber and Piston can cause higher compression than usual . Has you engine been Consuming Oil more recently or smoking during start up . A check of the Spark Plug and Piston Dome will give you evidence of excessive Black Carbon . Just a additional thought , since this could cause slow turn over during starter usage .

I do see smoke on start up, I'll check the plug--thanks