What would be the expected reaction in this case? We'd want (or require) those people to keep their desires to themselves and choose their employees/employers professionally without discrimination, right?
Thing is, you can't both have your cake and eat it; if we argue that companies should be allowed to act on political/social opinions, then that applies also if you don't like them; and if we argue that it should keep their nose out of what position their employees take on various political/social issues and just focus on their work, then again this will apply to both positions that we like and those we hate.
If employers are allowed to fire (or not hire) someone for pushing anti-LGBT agenda, they'd be allowed to fire (or not hire) someone for pushing a pro-LGBT agenda. Either employees have the right to freely express their political positions or they do not - but that right doesn't depend on which of opposing positions they hold.
Your example of the HR person who has to make decisions about medical issues obviously falls in a different category, but even in this case discretion could be assumed to be the default.
For example, a CEO might not want someone who is anti-LGBT representing their company in any way. On the flip side of that, I wouldn't want to work for a company where the leadership was anti-LGBT. Even if these people are otherwise good at their jobs, a person's morality does matter when deciding who to work with/for.
You might make the argument that no one at work should even know if anyone else is anti-LGBT or not, or argue that people should put those cares aside in the interests of a harmonious workplace, but I just don't think that's how (all) humans work.
There are practical considerations to this, too: say a trans employee has to deal with an anti-LGBT HR person regarding trans healthcare and the company's health plans. The HR person is going to be uncomfortable about that conversation, and the trans person isn't going to feel like their concerns are being taken seriously, or even feel safe having that conversation in the first place. That's... not okay.
> On a somewhat related note, I don't consider my colleagues my friends, or, God forbid, the workplace a 'family'.
Just like any other random group of people in the world, some of my colleagues are my friends who I socialize with outside work, and some (most) aren't.
(Agreed, though, the whole "company is family" nonsense needs to stop.)