Believe me, I wouldn't be surprised if that was the case, and I 100% think this is wrong.
My take here is that filling out a government form and having it be rejected because you didn't put required information that isn't stated as required into a box that the form doesn't have and getting denied/made to redo it is an extremely plausible scenario. In the case of a cis person being denied this way, it's a mundane bit of bureaucratic nonsense that nobody would blink an eye at.
The article states:
The law has been in place in some form for decades, though it’s rarely been used and usually arises in the context of candidates wishing to use a nickname.
The fact that this law has been identified as a real problem for trans people and that there is a quote in the article from the (Republican) governor saying "this is bad, we should fix it" strikes me as acknowledgement that this dumb rule is disproportionately affecting trans people and should be fixed.
We have a depressing number of real examples of malicious use of the law against trans people, so all I'm saying is that this one doesn't seem worth getting fired up about unless there is evidence of actual malicious intent here.