In the Standard Japanese vocabulary, the adjective ない replaces plain negative of ある (whose regular form should have been *あらない). It's the same phenomenon with the past form of "go" being "went" in English, called suppletion.

So the answer is: it is both an adjective and the negative of ある.

I did see 無い used in that way, and obviously the kanji made the distinction clear.

There's no distinction and whether they use kanji or not belongs to personal preferences. It's however true that the older the more kanji they tend to use, since the post-WWII mainstream orthography has discouraged people from using kanji for function words.

I'd mainly find this concerning for polite speech, since there's a more obvious difference between ありません and ないです.

This is a good point. Prescriptively this effect of suppletion only involves its plain form and the polite negative is regular: ありません.

However in reality, we do apply it to polite form and make a phrase ないです. The two forms are competing, and when to use which is another somewhat delicate problem.

Also see: 「ある」と「ない」は動詞か形容詞か…

In the Standard Japanese vocabulary, the adjective ない replaces plain negative of ある (whose regular form should have been *あらない). It's the same phenomenon with the past form of "go" being "went" in English, called suppletion.

So the answer is: it is both an adjective and the negative of ある.

I did see 無い used in that way, and obviously the kanji made the distinction clear.

There's no distinction and whether they use kanji or not belongs to personal preferences. It's however true that the older the more kanji they tend to use, since the post-WWII mainstream orthography has discouraged people from using kanji for function words.

I'd mainly find this concerning for polite speech, since there's a more obvious difference between ありません and ないです.

This is a good point. Prescriptively this effect of suppletion only involves its plain form and the polite negative is regular: ありません.

However in reality, we do apply it to polite form and make a phrase ないです. The two forms are competing, and when to use which is another somewhat delicate problem.

Also see: 「ある」と「ない」は動詞か形容詞か…