Welcome to the forum. Great to have another Gravely nut around.
Those are both very nice looking machines. Looks to me like they have been used, but taken care off too.
OK, now to the problem. It sounds to me like the timing might be an issue. I have had this happen a couple of times. When mine is a little out of time, it will start well on a cold start but not on a "hot" start.
Since the battery is, hopefully, ruled out, the first thing that I would do is install a new plug, wire, clean and set the points, and set the timing. The plug and wire together will run you around $20 if I remember correctly.
For parts, call up Richard over at Richard's Lawn and Garden. Here's his web site: http://www.gravelyparts.com
He is great to work with and is very fair, not to mention an encyclopedia of Gravely.
Good luck and let us know how all this works out.