I think for so many people, the 'issue' isn't an issue because they deem it to be scare mongering. Not only that, but it generally doesn't impinge on their lives so they have a very laissez faire attitude.
The moment their own children or environment is affected, then it's a different matter.
The principle of self-ID seems rational. If someone wants to identify as a particular gender, in the modern world of 'live and let live' many are happy to support the self-ID principle. Then, when the potential side-effects of self-ID are explained, I think they tend to be viewed as searching for a loophole that might create a theoretical threat. i.e. scaremongering.
It won't be until a number of abuses of self-ID occur that the general population grasps the issue and by then, there will be huge reluctance to overturn regulation, instead, it will most likely just be put down to 'crimes that could have happened regardless of self-ID' - because a handful of new opportunities afforded by self-ID will be absorbed into the general mass of sexual crimes committed by men. In some cases, it may even be put down as a female on female sexual assault which will further skew the reality.
Many people are misguidedly believing that fears over self-ID are about assuming transpeople are predators. That is not the assumption at all. The assumption is that some transpeople will be (just because statistically a percentage of both sexes will be), but also some predators will utilise self-ID for their own ends, which has nothing to do with being trans, merely about offering an opportunity for predators to breach what are normally a safe havens for females.
And of course, there is the issue that with perfectly legitimate transpeople, children risk being exposed to seeing the genitalia of the opposite sex. The pros and cons of that are not a trans issue, simply a matter of parental concern.