I've never put anything on our grass or aerated it anywhere I've lived and not had significant (I have had and still have one odd tiny patch) problems with moss.

Next door has problems with the strip of grass that passes our kitchen window. They put something on it and it all went black and looking horrible for months. Grass has come back but I notice moss has too.

The mossy area stops around the gate to the field which is where I start cutting (it's all part of the same farm track/right of way). It think they have a bit of a disadvantage in that their tall wooden fence makes it the most shaded part but I think they are cutting their bit way to short. By my own (perhaps faulty...) way of thinking, the grass and moss are in competition and they are giving the moss the upper hand.

I've got 2 mowers, a 55cm petrol one which has a lever to set to mulch and a small battery one which needs (and always in place here) a bit inserted to mulch. I must admit I don't think I've ever particularly thought about it benefiting the ground. As long as you don't let the grass get too long or cut wet when you will get clumps, it is also a hassle free way of having a tidy(ish) cut without the need to empty a mower's grass collection box.