I've noticed on Mumsnet that some people do like to wallow in misery / woe is me mentality.
For example, there was a woman who's son was choosing to soil himself at school, rather than use the toilets. He was apparently fully toilet trained at home. Various people made practical suggestions, such as trying to get the boy to move his bowel movements to the evening. That was immediately declared to be 'impossible', as were all the other suggestions.
I think 'mental load' is just an excuse to moan for some people, but a massive thing for others.
In our house, things are split fairly evenly. My wife refuses to drive because she doesn't like it and I get stuck with all lifts. I also do all the 'adulting' and my fault share of laundry, washing up, cooking, etc. I work full-time and my wife is still part-time due to young children. She obviously does the childcare while I'm working, but she's currently sleeping off a hangover and I've done swimming classes this morning and I'm now preparing the kids' lunch.
In contrast, a mum we know does absolutely everything. And I do mean everything. Her husband is out drinking after work most evenings. I expect she's a mum that would find her workload reduce if she was sans husband.
To summarise, I think you can hate the phrase, but it's very much real for some people - but it's also massively over egged by many.