You kind of answered yourself, he's a very good NBA player when healthy but that's almost never or he'd be starting on a contender making 25+ million per year. To put this in perspective, he was drafted in 2018 and since then he played 250 regular season games (out of 579 games, 43.2%) and 4984 minutes (out of 27,792, 17.9%). If we only take the 4 past seasons (including this one), he played 76 regular season games (out of 271 games, 28%) and 1524 minutes (out of 13,008, 11.7%). Those numbers are quite telling for a player of his abilities.
IMO the purpose of a third string center is to have him available when you need him to cover for your top guys, and if he's always injured that kind of defeats the purpose, so I'd aim for a 3rd string center who can cover for Wemby's own issues rather than one who makes you think of a 4th center to cover for his, there are going to be a lot of minutes available every season I'm afraid. And this doesn't even account for the 13 million a year he's making and the assets it'd take to acquire him, that may be better spent elsewhere.
I think there's got to be a solid option somewhere in the G League, undrafted, wherever, some teams have the staff to identify these guys and routinely do, it's a shame Spurs aren't among those teams.