The problem with stereotypes isn't that they have no basis in reality. The problem with stereotypes is that they are just as inaccurate as they are accurate. The problem with stereotypes is that they stunt the potential of tens/hundreds of millions of people.
Do we really want to live in a world where women are denied managerial/executive jobs, because of stereotypes that they can't control their emotions?
One where African Americans are rejected from job applications, because of whatever derogatory stereotypes people hold against them?
One where Jews are routinely judged as being immoral and obsessed with money?
One where Southerners are socially shunned for being uncultured racist bigots?
One where women refuse to date engineers because they're socially incompetent boring dorks?
One that bans Gays from being teachers because they're likely to be child molesters?
If you're someone who thinks that making life-changing judgements about individuals on the basis of stereotypes is perfectly fine, the early/mid 1900s are right up your alley. Thankfully, most of us have come to recognize how damaging and morally repugnant such a system is. I would hate to be judged and discriminated against on the basis of stereotypes, and hence, I refrain from doing the same to others. Ultimately, that's the courtesy that we as a society have decided we should extend to one another.