Let's look at the symmetrically opposite situation - for example, a CEO might not want someone who is pro-LGBT representing their company in any way. On the flip side of that, some people wouldn't want to work for a company where the leadership was pro-LGBT.

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.