I always thought resetting was something exclusive to the being with the most determination Underground. How does Asriel still have the ability to reset if he left the surface? Is it because the barrier is broken? Similar question, just to clarify Frisk and Chara would now be the only ones able to remember any resets done by Asriel, right?
True Resets exist, therefore it's not exclusive to being in the Underground.
The way I interpret it, lore-wise, is that the ability to save and load is a failsafe the seven mages added to the barrier in case a human ever fell down. A baked in survival mechanic- so long as they had the will to live, they'd be able to save themselves. They didn't account for nonhumans with determination. Basically, the barrier is what 'unlocks' the ability, so even when on the surface, the power could remain because the vast majority of humans simply never crossed the barrier.
What this also means is that no new humans will be able to obtain the ability going forward. Now that the barrier is gone, the power cannot be spread to anyone else.
In the end, this means that the only ones who would keep such a strong memory would be Frisk, Chara, Flowey/Asriel, Undyne, and now Mettaton thanks to the amalgamation giving him rippleproof memory too.
Granted, of them, only Frisk, Chara, and Flowey/Asriel are still on the board.