The US is taking a strong lead in brokering a deal. It got Ukraine and the EU to make large concessions (mineral deal, extra EU defense spending) to signal to the US that they are serious about peace and serious about heavily contributing to it to make it possible.
If Russia's public answer to this is: NO, I will not even lift a finger, then they open themselves up to US scrutiny because they admitted they are 100% unwilling.
As you rightfully put, they are already unwilling, but saying it is another thing. It's much better to go into a peace negotiation suggesting you're willing, than to block any talks at all.
What does US scrutiny look like? I have no idea. It could mean more sanctions, more weapon deliveries. Or it could mean nothing at all. I can't read this administration, it's so random.