Firstly, one may not braid hair on Shabbat regardless of how one categorises it. In the Mishna, Shabbat 10:6, braiding hair is forbidden by all authorities, the only difference being whether it is forbidden MiDe'Orayta or whether it is forbidden MiDeRabbanan. Rabbi Eliezer holds that it is forbidden MiDe'Orayta, and the gemara's interpretation is that this is because it falls under the category of "building". The chachamim disagree, but still forbid it because of shevut.
Secondly, there are innumerable instances in which the simple meaning of a verse is "subverted" in order to establish the halakha. There are principles, however, by which this is done. These principles, while they might not align with current systems of logic that you are familiar with, are nonetheless rigorous and (somewhat) consistent. They are known as middot, and if you take a little time to familiarise yourself with the principles of midrashic exegesis, you might be less surprised to see things like this.
In fact, the Mishna in Hagigah 1:8 acknowledges this problem when it declares that the laws of Shabbat "are like mountains suspended on a hair, for there's [only] a little bit of scripture, but a great many halakhot."