We do a mix of ToTM and grid... and no it's very much not the effect of TotM, not even seeing how that would matter. A Rogue with 35' movement can, using Cunning action as a dash movement move 110' total... if that's not enough to get out of the melee range of most creatures (especially if they are already engaged with another party member), not sure what is... and what exactly do you mean a specific reading of Cunning Action... it's pretty clear to me but what is the alternative?
You can Disengage OR Dash with Cunning Action, so you'd be mostly taking an OA every time you did as you suggest, and granting advantage on that OA because you're dashing IIRC. IF you can use Cunning Action during your split move, which isn't clear, then you can move up, attack, Disengage, and finish your 35' movement without being subjected to an OA. As I said, that's not a lot of distance you're putting, though I don't doubt that it can be enough in some cases.
How can a Rogue using the Mobile feat and Cunning Action not get away from the creature? It gets no OppAttk on him and he can dash as a bonus action on top of his regular movement...
I don't know about using Mobile. The rules lawyer in me says there's a difference between a 'Dash Action', and 'Cunning Action', which lets you 'Dash', but I think RAI likely is Mobile gives you super kiting. This is another thing with the TotM Effect, it sounds plausible until you put it on a map, at which point the PC is running 70' up to an enemy (no stealth involved) hitting said enemy, and running 70' away again, and there's zero chance of any retaliation? Well, its a fantasy game, but that's some serious kiting. 5e really has some pretty loopy combat rules when you start looking closely at them.
In any case, our rogue player has certainly not considered this, she's 79 and is a great player, but really has no interest in being that into tactics. She's got a 20 DEX and IS quite capable of putting some hurt on things, but what goes around comes around.
I'd also just say that in general monsters often do some pretty hefty damage in 5e. Not as much as PCs, but it is not at all unusual to have 80% of your hit points chomped off in a single hit. And monsters have tricks too, so its not like you can just run around kiting everything all day and being unscathed.