You've completely misunderstood what "mental load" means haven't you? As have a lot of people.
A "bored SAHM" (if there is such a thing) wouldn't be experiencing mental load in the first place. Mental load comes from doing more of the share of household tasks than the other partner, shifting the burden unfairly. That doesn't really apply in the case of a SAHM.
If you have a clear division of labour with a breadwinner and a homemaker it's less of an issue (it brings all sorts of other problems but that's a different thread). Because people focus on their respective "jobs".
The problem comes precisely when both partners are working full time but only one partner (almost invariably the woman) is doing all the strategy/planning/anticipating and the other one is only doing the work and then a few other things handed out to him but without thinking about how this impacts no the rest of the household.
A good example is a husband who agrees to go on a work drinks event when he knows there is a parents evening coming up, fails to check the family calendar and thus puts his wife in a double whammy: 1. she's doing the parents evening alone (again) and 2. she will have to source childcare because the husband is out on the piss with colleagues.
Or, the husband who arranges for the whole household to go around to friends' houses for a barbecue without taking the trouble to check that one of the kids has soccer practice/drama club.
People always say "it's easy to do this stuff, it's just adult life", and talk about how easy it is to pay bills by direct debit etc etc...but they are missing the point. It isn't the ability to do the stuff that's the problem, its when you fail to anticipate that your needs may clash with those of the rest of your household.