Puckett is certainly a viable choice. He played plenty of one pocket.
Beard
U.J. Puckett frequented a hole-in-the-wall poolroom called "The Golden Nugget" in the early 70s. It was on the west side of Fort Worth on Seventh Street where it joined with Camp Bowie Boulevard, originally set up by the senior Mr. Cecora and later his fat cheery braggart snake-oil salesman son Gary, who somehow had snagged the most beautiful wife in the world. Her name was Carol, a tall, dark-haired, buxom and better-looking version of Cher, and there were stories... but I was poor and very married to a pretty good looking chick myself.
Puckett would show up regular like and brighten up the crowd. Just his wild white-hair and smiling presence was enough to get the place a-rocking. I've never seen anyone who could always say the right thing to get a laugh, and at the same time take the air out of someone too big for their britches, and without making it too painful. He wore a straw hat often at that time, wild white hair poking out everywhere under the crown, but about then or shortly after, I first saw him in a tall white round-top cowboy hat that became sort of a signature.
We all called him Puckett or the railbirds Mr. Puckett, few at The Nugget called him U.J. at least when I was there. He never ever played anyone or even seemed to gamble at all. When someone offered a game he would break out his great big smile and drawl "Awwww... Ah cahn't puh-lay..." but every once in awhile he would put his old steel-joint cue together and make a good stroke or two on a table before they put the balls away.
A few older guys were regulars, especially the week after their SS check came in. Magnolia Red, whose real name I forget (edit: J.R Richardson was his name, as I now recall), was one of the most colorful and apparently a long-time friend of Puckett. He put his gin or vodka in a silly looking pink Johnson's Baby Oil plastic squeeze bottle, the one with a tiny hole in the spout. He would throw his head back and squeeze a big stream of alcohol down his throat and then start singing off-key at the top of his lungs. His usual game was one-pocket against senior guys his own age for modest stakes. But I remember one time he had a game on a little 7-foot Brunswick table, eight ball I think, with a local Fort Worth hotshot (and he was a killer player most of the time) and his backer, maybe getting some sort of spot I don't recall exactly. Red got drunker and drunker as the set went on, singing louder and louder and acting stupid and making his opponent mad, and the opponent would proceed to miss and dog it time after time. So then Red wins the session for a big $2500. The next day rumor had it that Red had filled his little pink Johnson's Baby Oil squeeze bottle with mostly tap water, and never was the least bit drunk at all.
I saw Puckett often around the Ft. Worth pool scene those 70s years. He migrated to an east side pool room long since closed, which had a "U.J. Puckett Room" glassed off supposedly I guess for action play but maybe it was just the non-smoking room.
I would like to see U.J. Puckett in the HOF for being an ambassador and entertainer to everyone at the poolroom, not just the gamblers and players but the regular working folks who never bet above two dollars. I never saw him play for money, yet I have great memories of a few encounters with a formerly great player who, though well past his playing prime, was a huge attraction, and rightfully so, every time he entered a pool hall.