The word "foist" does imply coercion to do something that is against the target's will or interest. Would you be equally okay with the statement being replaced with "self-sacrificial status climbing is more often foisted upon men in our society"? Add to that the common narrative* that men are in charge in our society and therefore social pressure is essentially equivalent to pressure exerted by men, and you quickly get a statement that amounts to "[the problems we observe] are caused by men forcing women to care for family members".
* Even if you don't actually subscribe to this position yourself, it's unfortunately in the nature of debate on a politicised topic like this that the moment you take a position associated with one of the standard camps ("the American Left" or "the Blue Tribe" here, I guess), people will assume that you also hold other opinions typical of that camp. Certainly, "men hold the power in American society" is one of those typical opinions.
* Even if you don't actually subscribe to this position yourself, it's unfortunately in the nature of debate on a politicised topic like this that the moment you take a position associated with one of the standard camps ("the American Left" or "the Blue Tribe" here, I guess), people will assume that you also hold other opinions typical of that camp. Certainly, "men hold the power in American society" is one of those typical opinions.