It really doesn't. The front in eastern Ukraine has been shelled into oblivion at this point and it hasn't made it easy for Russia to advance.
That's because Russia is also short on shells. The bulk of their advance in April-July was facilitated by their mass artillery fire.
I'm not sure that's true. Operating in a radioactive environment makes every single task so much harder. When Russia can barely afford the time to teach its recruits how to shoot, how could they possibly spare the time to teach them how to manage radioactive contamination?
In this case "nobody can have it" is effectively the same as "we have it" in terms of keeping NATO at a further distance from the important parts of the federation. Or whatever the Kremlin justification is these days.
Is Russia low on land? Are they running out, that depopulating several nations and holding their territory with conscripts makes strategic sense?
In the hypothetical of them using nukes, we'd probably be past the point of there being any reason other than to wipe out the other side and show force to the rest of the world. That they control/own it by might sheerly for the sake of doing so and not by any other rationale.