I always thought that the other kids' runs could have been tampered with via Flowey. I headcannon that, due to their souls main attributes being not of determination, that would leave Flowey still in control of SAVE, LOAD, and RESET, and ultimately the most determined person in the Underground. I always thought that was just him messing with the timeline while the kids were there. But that's just me.
I’m fairly certain that Flowey was created after the other six fell. His end game is getting to the souls in the jars, and he was never able to do that until you arrived.
The game also strongly implies that only a short time has passed since Flowey and the Amalgamates were created by Alphys before Frisk falls and takes over the Resets. Most monsters don’t recognize a human, so it seems to have been a long time since the last one fell down.
The other kids may have been able to save and load to some degree based on comments by Toriel and Alphys’ analysis of their souls. But I guess since their primary attribute wasn’t determination, they eventually gave up against Asgore. Meanwhile Frisk keeps on truckin’.