"What's particularly dated about Falsettoland?"
As Rent put it: people living with, not dying from, AIDS.
The first part (March of the Falsettos) was about relationships. Finn was pushed to create an "ending" to that show and he wrote Falsettoland. However, Falsettoland became less about relationships and more about social issues. The opening reads like a Democratic Party platform "Homosexuals, women and children..."
Everyone completely ignores Part 1 (In Trousers) which has some better songs and is in many ways superior to Falsettoland.
IMO, Falsettoland is the weakest of the trilogy because it moves away from the universal subject of relationships and tries to be a musical The Normal Heart.
If anyone ever tells you that you put too much Parmesan cheese on your pasta, stop talking to them. You don't need that kind of negativity in your life.