‘Of course’, it does lead to the obvious absurdity that a woman who fulfills all the good qualities of the poem is a man! “The Female Man”, perhaps...
If you can't make a trivial fix to a simple problem, you aren't paying attention to the poem's message.
It would be like, in the pursuit of lowering your sodium intake, making bread without salt. Is it healthier? Maybe. What does it taste like now? Cardboard.
But...it doesn't. While the fictious direct addressee is the “son” (whether literal or in the figurative sense of an protege of an elder mentor is neither explicit nor important, since the whole impact of the poem is in its general applicability not in it being specific to an addressee) of the speaker from whose POV the poem is told, the son/man pairing necessarily states that the qualities are tied to adulthood/maturity, but is just as valid if they are not inherently tied to gender as if they are.
Having grown up as the child of a single mother, I don't think many men ever stop to think where we'd be without them. It's not a place I'd care to exist.
Most men are only as successful as they are because they had a loyal and dependable partner at their side to help propel them there - if by nothing more than taking care of all the shit they have no time for while they're so busy chasing their success. I wonder how many men would be half as successful without their partners supporting them and encouraging them forward.