I think the distinction between "moved" and "ran" are important here. Scholes could move all day and had no problem with being perpetually open due to his subtle moving around constantly working for space, but 'ran plenty himself' would need some context because when we talk about running now, we're discussing drones getting up and down the pitch in a number of the highest intensity sprints that have become so paramount in the modern game. Scholes categorically did not have to sprint repeatedly game in and game out. In fact, it's that high intensity H.I.I.T. work that is ruinous for the throwback players who are skilled but nothing like the athletes who can perform all that high octane stuff.
I think Scholes had more stamina than Kobbie myself, but the conditions set via the quality of their respective teams, and age of them as well as the way we used to dictate the pace more often than not, makes it much harder to assess. And as above, Scholes would be ruined in a high octane environment where he doesn't get to "move" more than have the readiness for yet another lung-busting sprint 40 yards before having to turn and be ready to do it again in the blink of an eye. This is where Mainoo visibly flags, but outside of that, he doesn't look much flustered, imo - it's the moment the game turns into an athletics meet that he is found wanting and absolutely bolloxed within 60 minutes.
It's interesting that you mention Beckham, who was famed for his ability to complete the beep test - put him alongside Keane and also have the absurdly athletic Giggs completing the midfield unit and it's no wonder Scholes got to pick and choose his moments. If only Mainoo had the equivalent of such talented yet supreme athletic specimens to play alongside; perhaps then he wouldn't have to do so much H.I.I.T. himself!
An easy way to assess these two is in trying to imagine what the other would look like in a team switch. Does Scholes struggle if asked to play Mainoo did/does in a terrible set up under ETH, or with so much on his plate athletically in an Amorim team? How does Mainoo do if tasked with Scholes' jobs in an all-time midfield unit?