You do not invent your own name sign. Name signs may only be given by a person in the Deaf community. Some hearing people (like interpreters and teachers) mistakenly give name signs without realizing they are in violation of Deaf culture traditions. However, a name sign cannot be assigned by a hearing person.

American Sign Language has deep cultural and linguistic significance. Typically, it is not until you are involved in the community that you are given a name sign. In fact, not everyone within the Deaf community has a name sign.

From article
It's like a cultural thing. This still doesn't really answer why it's like that in the first place, but I think in general the reason it feels inappropriate to name yourself in another language is that it feels "cringe" for lack of a better word. Somebody picks a name that has all kinds of cultural and colloquial associations without understanding them at all.
That's kinda my theory of cultural appropriation; it's not wrong because of some deep ethical reason, but rather it's just often uncool. People sending signals that they don't understand themselves.