If you’re into breaking things, I’ve noticed lately that a lot of consumer DVD players have such an assembly on their front panels (usually tact switches behind them I guess). The ones I’m looking at here range in size from about 3/4" diameter, up to over an inch and a half, and look as if the cap part is just sitting in a simple round hole in the panel. They’re mostly 4-way, but I do see a Sony here with that center button you describe (a DVP-NS300).
Once the cheap loader assembly starts screwing up in most of those things, they’re worth about as much as that switch. The earlier players that refuse to play anything are worth that much from the start. 8)