Would this be seen as a genuine gesture or as performative marketing? Could it have an impact—positive or negative—on how people perceive the company?
If you really want to do something good for X group of people... just make sure your business runs well and treats EVERYONE properly. That's it.
No need to advertise it. Work on the core aspect of being good and doing your job right. That's what everyone wants anyway.
If you want to support the gay community, support the gay community in ways that actually matter. A rainbow on a logo doesn't matter.
Once you start adding other logos, they will bring associations that you cannot control. This will dilute your brand.
Think about when you want to signal inclusiveness (eg, in employer branding) and highlight it there. Then it will become part of your general brand without the need to give up your distinct logo.
I have no problem with "gay", "lesbian", "bisexual", "transexual" and other identities, but I don't like LGBTQIA+ (or whatever it is today) as a label because I believe that label erases the individual identities involved. In fact, labels in general tend to obscure reality, see [1] [2] [3].
As an activist I've seen that the strategy of taking the most radical position on every issue and putting up the tallest flag just doesn't work -- people get disengaged and stop going to your protests.
When I saw "I" added to that list I was hoping it was "Incel" (talk about people who "can't be heterosexual") but, no, it is Intersex -- some Intersex people feel violated by the same surgeries that help transgender people feel whole.
I was annoyed, as a foxworker, that "two spirits" was added to this list without consulting me. I certainly do contact female fox spirits but I also contact male fox spirits too and the gender of them doesn't concern me so much as the interspecies difference. I don't see any reason why I should buy into the reductivist gender paradigm of the day or dislike the Harry Potter books any more than I did when they came out.
[1] https://en.wikipedia.org/wiki/Bisexual_erasure
If you are going to add a Pride flag to your logo, the traditional rainbow one would be less controversial.