Yes, non-binary people will be able to choose between whichever gender's services they feel will benefit them best at any given time. They would always have the choice.
So I'm guessing all non-binary people will choose to compete in women's sports.
All non-binary people will choose to go to a women's prison.
All non-binary people will choose a female ward in hospital.
Note that the TRAs are not pushing for non-binary spaces at all. They are not lobbying for non-binary sports, prisons, hospital wards, refuges. They just intend to take over the women's.
Male non-binary people will still of course join male-only golf clubs, still serve in religious roles reserved for men, still choose to inherit titles, still put themselves as father on birth certificates, still claim paternity leave. Until such time as the law is changed and male non-binary people can claim maternity leave.
And then when these men are ill or go to prison, they'll switch to using the female services.
I'll bet we'll see male non-binary police officers refusing to search male suspects, male non-binary prison officers refusing to do a strip-search of a male inmate. Because these things are icky and dangerous and if you can get away with claiming that you can't do it because of your gender identity, then a lot of people will.