The brink of time is an incredibly better definition for it. That place is not where time ends, it's not the far far far future when universe implodes, or when golf becomes a good sport.
n. brink (brngk)
1. a. The upper edge of a steep or vertical slope: the brink of a cliff.
b. The margin of land bordering a body of water.
2. The point at which something is likely to begin; the verge.
Instead, the brink of time is a place outside of time, judging from facts, even a pocket dimension. Gaspar is not affected by the changes in time that Crono and the "other travelers" make, he seems to be able to watch all the changes in time from that place (Crimson Echoes Belthasar would be jealous), since he's always on date to the changes you've done in your travels. Gaspar lies on the very verge of time, on the corners of time, just far enough to remain unaffected by time travel changes.
But back on topic, this doesn't mean that time doesn't run there, Ayla still gets bored if you don't get her in party, Spekkio tells you that's it's been long since last seeing you, Gaspar takes a nap and wakes up. The time there is unaffected by the "central" timeline, which is why it's the brink... but there's still time going on in there. So time hasn't ended, so no end of time.
![Razz :P](https://www.chronocompendium.com/Forums/Smileys/default/icon_razz.gif)
To sum it up, the end of time means "somewhere" in time, the place being the end. The brink of time means outside of time, on the verge, the edge, the outskirts of time, where time has no reach, but still has a kind of time on its own (else the game would end after 2300 A.D., lawl). Just imagine time as a road, all srs business happen on the road as the road goes on, the brink of time also runs as long as the road of time goes... but it's right outside the road, so srs business don't reach there.