In Shakespeare's time, all women were played by men. A woman could not go to a theater and see a real woman on stage, only a man trying to portray a woman. Mind you the goal of acting at the time was not naturalism, but the effect was that an oppressed group was only visible through the interpretation of the dominant group. Some of those men may have been sympathetic to the female characters. They almost all knew at least some women. But imagine how it felt to only ever see a caricature of yourself on stage, no matter how well intentioned. The men on stage could go home and lead their lives free of the discrimination that pervaded their culture and that women could not escape.
That is the problem with casting straight cisgendered actors. No matter how empathetic they may seem, they are part of a society that is actively discriminating against queer, trans, and other marginalized groups. Groups who seldom if ever see themselves truly represented on stage or screen, and when they do usually only in niche productions. It is as harmful as blackface in that the practice of casting cisgendered actors in transgendered roles is another example of a dominant group blocking participation by the people who the role portrays. Individual performers may be well intentioned, but they are participating in systematic discrimination against marginalized communities.