They truly believe that any gay symbolism is pushy.
It is in your face, especially during pride month. But that's the point. It's in your face so we can talk about how gay people are treated, what they've overcome, and show that they are being included in places they previously weren't. You can't do that if it's not in your face.
So, they've got a point when they claim that gay symbolism is pushy. I'd ask them why they think it's useful to be pushy.
Why would they have to keep up with their appraisal? There is no downside to not doing it, other than not being worth as much if they want to sell or IPO, but that would be true even without the equity firm's investment and appraisal. No law says they have to meet the expectations set by a single investor. There is no legal duty requiring a company to maximize profits or shareholder value.
However, even if such a law requiring them to maximize profits existed, it's very reasonable that they would legally continue making the same types of movies that earned them that value in the first place. There couldn't possibly be a requirement for them to change business strategies, else every company would eventually all end up in the same, most profitable industry. They'd all be selling movie theatre popcorn or something.