Private is probably not the correct word, but flying the LGBTQ+ flag is probably a for of capital ‘S’ Speech, i.e. the thing the 1st amendement of the US Bill of rights is supposed to protect.

To what extent should government institutions engage is Speech is a question without an easy answer.

In this case the LGBTQ+ flag represents, in part, common sense civil liberties and protections for a community. Unequivocally a good thing, and to say otherwise is bigotry.

However the undeniable goodness of the Speech does not necessarily mean it is a type of speech we want government institutions to engage in. One method to illustrate this is to replace the clearly positive flag with a clearly negative one, say a nazi flag.

I’d expect most people here would have a problem with the dmv flying a nazi flag.

So we simply say that government institutions can only fly good flags. The problem is someone has to decide which flags are good. It may seem obvious which is which, but unless we put it to a vote, we’ll need a committee or a single person to make the call. And some people are evil, and would falsely claim the Nazi flag is the good one, and now we’re in a bad spot again.

So like all things there’s a lot of annoying nuance to be dealt with and sucks when it should be easy to just allow good supportive speech.