We see in-game that characters don’t forget their original pasts. For instance, Magus seems to remember a past in Zeal where the heroes of the game weren’t on the scene as we see in his flashback on the North Cape. Why should Crono be any different? Lucca also seems to remember a past where her mother was paralyzed by that machine, or at least that’s what I inferred.
If not, the pendant could have protected Crono from forgetting. We see in Chrono Cross that Kid’s pendant is capable of rewinding time to an earlier point to keep her from dying. There may have been residual energy in Marle’s pendant from the Mammon Machine – not enough to open doors or chests with crests on them, nor enough to rewind time, but enough to protect its wearer from endangering their own timeline or forgetting certain things.
The Epoch might not protect our characters the same way the gate key could. Belthasar is so far from the past that anything he does in 2400 AD or 1000 AD won’t affect him. As a result he can use the Epoch or the Chrono Trigger willy-nilly with nary a consequence to his personal timeline. By the time Crono and company get the Epoch, they’re still carrying the gate key with them. The Epoch doesn't need to grant the same immunity.
Thinking on it, Belthasar’s original plan may have been to return back to his home time via the Epoch, get the Chrono Trigger from Gaspar, return to 2400 AD, and use it on Death Peak to rescue Schala. Belthasar’s machinations would have to be very precise to keep the timeline intact and be able to rescue Schala. Unfortunately, that version of him died before he could complete his objective. The new version had to deal with a significantly altered timeline thanks to the intervention of Crono and company, and had to reformulate his plan. Perhaps the new Belthasar did go to 12,000 BC first when he completed the neo-Epoch, but found that the Chrono Trigger wasn't completed yet and couldn't be used, and that's where the ridiculously complex plan of Chrono Cross came from.