The simple answer to 'how their (asserted) needs impinge on the lives of others' is that there will be no such impingement. The legal recognition system will create no new obligations on private citizens. Re the government, they will no longer be able to treat trans women the same as cis men at all times. They will need to recognize and accomodate the special needs of trans women. For example, the rules of government programs, including health care (e.g. Medicaid) will need to recognize this difference, which would prevent DeSantis from doing what he's done to take away people's health care. But it will not cause government to 'step in' to change the behavior of private citizens. A gender neutral medical rights bill, which would not recognize that trans women are different from cis men, could not offer such protection, because hormones are not 'essential' for biological men in general, which means it is generally up to the government to decide if it is covered.

What I envision is that trans individuals will be given a new status in law, by opting into legal recognition. This will mean that the government must consider them a category separately from cis men in law. However, the law that creates this new category could also state that it will not create any new obligations on private citizens, nor will it affect the operation of sports and single-sex spaces. I can't see any downside to this proposal, honestly.

I've answered some of your other questions in my other response.