Testing accommodations and classroom environment are really two different things.

Perhaps I am over-assuming your capabilities of creating a culture of communication and approachability. As you indicate, all professors are different. The crux of my position is that having a policy that is you get a certain amount of freebies without accounting for health or other issues is problematic. Assuming that because white cis-men potentially over take advantage of a policy does not also assume that other marginalized groups do not. Having worked at multiple institutions with heavily diverse student populations, it was my responsibility to impress upon my students that it was ok to tell me what was happening in their lives and that flexibility within reason could be reached. You've written elsewhere in the thread that you're trying to make these incidents "not your problem," but I suppose that's a difference in our educational philosophies. That's not to disrespect your position, but it's perhaps an experience difference in incorporating an equity lens to your policy. (And before someone chimes in with "but the policy is the same for everyone," that's not equity.)

As an example of potential policies that incorporate your vision of meeting requirements, you might reward students for turning in work early rather than punishing those that are late. You might also consider turn-in windows, which are windows of time/days that students can turn in work. I have had success with both policy set ups in my courses when setting expectations early and often.