Wading into divisive, minority issues, seems foolish for a world leader unless the purpose is to sow division.
One person's divisive, minority issue is another's fight for the fundamental right to be treated equally under the law.
I'm not so sure. It kinda seems required sometimes if you're going to keep your coalition together. For instance, there are some people who are part of the Democratic Party base, who of all the issues out there, care the most about gender and bathrooms (I know some of them). Their stridency has inflected a good fraction of the rest of the party.
Unless you are an empire and want to oppress the majority…
No? A coalition is A + B + C + ... ~= 50.1%, you might need that vocal minority to make the math work. And those factions won't agree on everything, so there will be intra-coalition tension, but they know they have to work together to get what they each want. However, there are limits, so if the coalition totally abandons C's issues (or even just severely de-prioritizes them), then C will leave.