You've missed a few points in sequence. The first one is that these models are not trying to assess writing quality, they are not grading tools, and they are not intended to judge people on their grammar. They are intended to tell the user whether the text was generated by AI. The fact that they have a significant amount of false positives indicate that these programs are flawed, and it is likely that this is not the only flaw. For example, the big AIs are generally pretty good at grammar now. That wasn't true with the previous generations, and it's not always true with the small ones you can download and run on your own system, but if you've seen the output of GPT4, you should know that, while the content is likely riddled with errors, they will be presented in a grammatically correct statement. A program that tends to mark worse grammar as AI generated may also grade the better grammar of a LMM as human-generated. That's not the only possibility of how this demonstrated failure could mean that the program is useless.
Then, you equate the false positive to a way of indicating competency in English. It doesn't mean that, and the result will not be used as that. A person who fails to make their writing understood will be marked lower. A person who fails an AI check may be expelled for cheating. As I've already pointed out, it's not guaranteed that the only people marked as AI-generated by this program are those who don't speak English well, who I assume you don't care about, so what do you think about the problems that will affect others who are also falsely accused of using an LMM?
Then, we come to the assumption that people will simply not enter an institution where they don't speak the language perfectly. Observation is sufficient to indicate that this does, in fact, happen. They may do it in part to improve their language skills as well as studying the content they would want to. There are some subjects where specific languages are needed. For example, studying computer science is more likely to go well if you study it in English, because English is far more common than other languages in documentation for the tools they will use. There are books on programming in basically any language, but when you want to look up the library docs, an open-source project's readme, or even some larger projects' updated information, you may find that English is the only option. I'll also point out that tools to attempt to recognize AI writing are also used on researchers' papers to try to catch dishonesty. A lot of researchers will publish in the language spoken by many colleagues, which for most of them is one of a few linguas franca, English common among them. I've read many papers written by non-native English speakers, and sometimes this results in different writing styles, but since they have been reviewed a lot of times, these style differences don't often prevent understanding. Punishing them based on an erroneous program, sold by some company who couldn't be bothered to test that their tool worked and who probably lied about its accuracy, is harmful and outweighs any potential benefit of the correct answers that program might at some point generate.