I guess it is a luxury but one for us both - and it is the WOHP not often acknowledged to benefit (unless the WOHP is the mother - then it tends to be 'what a great guy, staying home so she can pursue her career')
For my DH, who is the WOHP, the benefit is in part because of all the logistics I, the SAHP, do with the DC - school, nursery, clubs, school trips, sickness, doctors, dentists, play dates, parties etc etc, and all the 'stuff' that goes with each of these things - but also all the emotional side of things with the DCs. My DH pretty much leaves it to me to deal with (or make any observations of) possible upsets the DCs might be having - settling into school, issues with friends/ teachers, school work, handling new situations and feelings, coping generally with life.
Sure he will contribute, and I know he cares - but safe in the knowledge that I will sort it all out and ultimately the responsibility is with me to manage everything and everyone in terms of general health, happiness and wellbeing.
To be honest this sometimes feels like a tremendous pressure, whereas going off to work, even when in a pressured job, could feel like light relief by comparison! Even more so if you know they are with the other person who loves them the most in the world.
It's not just the physically being there, it's the parenting and 'upbringing' you get left with.
That's not to say I don't appreciate and acknowledge that he is a good provider and we are privileged to have the choice, it is just there are so many subtle, less visible elements of the 'load'.