It has me worried. I've known 'biological women' who have been assaulted and attacked because someone else viewed them as too masculine to be women going into a women' toilet, this type of law seems like it would just encourage more violence by the general public and the police against anyone not toeing public perceptions of binary gender.
Sadly as much as I'd like to say otherwise, I think it will gain wider support purely because of where the US is at at the moment. As much as I would love this to encourage more gender neutral single person restrooms cubicles, I think the mentality is going the other way in a lot of places.
And that's before we get into the issues with that portion of the law that are more complicated than keeping those who haven't legally changed their gender marker on their paperwork -- mainly because it is the US and the laws between states vary so much and it has a big impact.
For starters, there are some state where it is never legal to change ones gender markers no matter the hormones, surgery, medical paperwork, and so on. I was born in one of them. In fact, some people born and raised in those states and moved to others have had roadblocks put in the way of doing so because not being able to change the markers on paperwork in one state can make things applied for in other states invalid because they do not match up. Things like medical insurance. This pretty much stops people getting legal changes because it is putting their health at risk - especially if they have any kind of disability or need for long term treatments or care. And then there is the financial roadblocks in states that do allow it but require a court order to do so - it's quite expensive, especially if you have to travel back to that state to get things lined up when you now live in another and/or have to do it in multiple states. And that's just the tip of the iceberg of barriers.
Getting a gender marker legally changed in the States, especially if one has moved around at all, is far more complicated than it is in the UK. So this isn't just a matter that people will be okay because all the real ones will have gotten their paperwork dealt with, literally if they were born in the wrong state they could now have serious issues as will those who are cis/'biological' who someone else decides isn't enough of their gender to not be challenged on it which, in my experience and the research I've seen, is more likely to be people in groups that already facing more public and police violence.