I think I'd install a thrust washer between the gear and the distributor housing to minimize the vertical play. A few thousandths is OK- - - -visible vertical movement isn't. We even used to install a "thrust button" between the timing cover and the front end of the camshaft to avoid timing changes from the camshaft moving fore and aft on the race engines, and that was not nearly as much timing change as the vertical play of the distributor would cause. Good detective work, BTW!
Jerry

"It is better to be silent and be thought a fool than to speak and eliminate all doubt!" - Abraham Lincoln
Cringe and wail in fear, Eloi- - - - -we Morlocks are on the hunt!
There is nothing noble in being superior to your fellow man; true nobility is being superior to your former self. - Ernest Hemingway
Love your enemies and drive 'em nuts!