I think that it is a difficult and emotive subject because, ultimately, being a SAHP and, especially a SAHM relies on a model of marriage which frequently does not work out.
Clearly (and it should go without saying, but this is MN) what I post is my opinion and does not apply to everyone. There are brilliant relationships with a SAHP where it works well for everyone.
I think it is really good for a child for a parent to be around a good deal of the time in the pre nursery years (I.e up to about 3 years old). This is a career interruption and should not prevent a return to work.
With school age children, there start to be meaningful issues. Firstly, if you ‘stay at home’ you need to take on a housekeeper/organiser role for both your husband/wife and children. Many resent this and you see the term ‘wife work’ bandied around. However, if you are at home all day, what else are you (materially) contributing to the marriage?
Secondly, there is the thorny issue of divorce. What tends to end up happening is both parties are resentful as the WOHP parent feels that they have been working all hours for their children, whom they will now see part time, but will still have to pay for full time (fair, given the circumstances, but it is the circumstances that I am questioning). The SAHP, in the others hand, often feels forced back into a workplace where their skills and confidence have deteriorated, and where they command far less respect than had they carried on working.
So, the SAHP model works well for some but carries considerable risks.