I think this is true to some extent, obviously it's a generalization.
Although - it might be that we could reframe it, and say that in middle class households, more women take on a male pattern of working and paying for daily childcare, and then both mother and father take a turn at off-hours childcare.
Whereas in wc households I think it's been more common for women to take a break from paid employment when kids are younger, as the wages don't cover childcare as well, and also in some cases they may see work more as a means to an end rather than a career and be very happy to see being a SAHM as a calling, at least for a few years.
There also could be something of a skills differernce. Among MC people, a lot of the jobs are pretty evenly divided among the sexes, and they may be more likely to also pay for other household work like repairs or yard work. Leaving both to manage cleaning and children.
It's not unusual in wc homes to see the employment as more divided by sex as well - women in more traditionally female jobs (care work, including childcare, shop assistants, cleaners, secretarial,) and men are more likely to be in trades. And you see the same division of labour at home - the mother may well be managing the finances, cleaning, household admin, and childcare, while the father takes care of household repairs, yard maintenance, can care, and other heavy jobs.