In theory, you are probably right. But a switch to self ID would make the rules/laws that you say don’t change kind of unenforceable, does it not?
Which rules/laws specifically?
Most of the law is untested in court AFAIK.
Technically the answer to your question, I believe, is no, changes to the GRA shouldn't make a difference to matters relating to the Equality Act.
The grey area seems to be around interpretations of the Equality Act as it stands and exactly what rights that extends and to whom. But since those rights don't explicitly require a gender recognition certificate to obtain its hard to argue that changes to the process for getting one of those certificates would change the rights offered under that Act.
Enforcement is a different kettle of fish of course, but again, legally it already is. I think you also have to distinguish between matters of policy and individual actions.
It's not illegal to challenge someone who 'appears male' in the ladies' room. It probably would be to have a policy that everyone who 'appears male' in the ladies room should be challenged as a matter of course.
Of course, as it stands, it is pretty much impossible to tell without more indepth knowledge of an individual whether they are transitioning or not but that doesn't mean that saying 'I am a woman' makes it so. In theory, it's also no different for a 'manly' looking cis-woman or, as others have noted, a transman using the ladies' room.
This being the case, my personal conclusion is that the best way to deal with it is simply to accept people using whatever facility they prefer on the understanding that there are standards of behaviour expected of anyone regardless of gender and that people acting in an 'iffy' manner are free to be challenged/ejected.