No.
Why is it so important to you that sex/gender is binary? Maybe try to relax and be a little less judgemental. It really does not matter for you what other people's sex is.
Intuitively, for normal people on the street, and arguably for politicians trying to define that sex is binary, how you would read a person is their sex. So that is, I would say, the historic traditional way of determining sex: by look and intuition (called common sense, usually). That's also how sex is determined at birth. (Note that in some cases, nurses need help with that, because it's not intuitively clear...)
Because this intuitional method does not work well, other ways have been tried to find something more objective than having a quick look. We found the sex chromosomes. But they are not binary either, as mentioned above. Now, one way of making chromosomes forcibly binary is to take the check for a Y cromosome, as you say. But that's just one arbitrary simplification. It is not an universally agreed criterion, because it does not always match with other criteria, particularly looks/intuition nor hormones.
And even intuitively, I argue, you should know that sex is non-binary. There are people where you cannot easily read the sex. You have an immediate intuition most of the time, sure, but not always. And then, as mentioned, your intuition may contradict chromosomes and/or hormone level categorisation of that person.
Another example: typically, an XY chromosome person with female external genitals is most probably read as clearly female. But by chromosome judgement, as you suggest, they would be clearly XY male. And at birth, they are most probably be classified as female. But they do have testicals and high testosterone levels, which again is totally in line with the male chromosomes.
Sex is not binary. Really. The world is not that simple. I'd argue that instead of trying to force sex to be binary, we should just try to not care so much about sex and gender. Because why?