Demonyms don’t use the same rules as countable nouns. Both “French” and “British” are acceptable demonyms, they’re just not particularly idiomatic in American English (which likes to overcorrect with “person” like you’ve noted).
(There’s no particularly consistency with this, it’s just what sounds “good” to American ears. We’re perfectly fine with “as a German” or “as a Lithuanian.”)
Precisely so: the OED's role is descriptive, in that it is to describe English as it is used, not how it ought to be used. It provides evidence of a grammatical rule - it is not the rule itself.
You can speak however you like, there is no language police, but the fact is the average English speaker will perceive certain constructs to be grammatically incorrect. "He eat", "I driving", "a French" etc.
(There’s no particularly consistency with this, it’s just what sounds “good” to American ears. We’re perfectly fine with “as a German” or “as a Lithuanian.”)