All true written languages are phonetic to some extent, even though they may not be alphabetical the way English is. Chinese characters have phonetic components indicating tone and pronunciation, Egyptian hieroglyphs are largely syllabic, Norse runes are alphabetic like English. In Chinese and Egyptian, there are purely non-phonetic symbols representing ideas (and other things like determinatives), but most have some kind of phonetic meaning (this is my understanding at least).

There's a spectrum of how phonetic a language is, where Finnish is on one end (sounds very closely align with spelling) and Chinese characters on the other, but all written languages are phonetic to some degree.

Norse runes are just an alphabet. As long as your language uses the same set of sounds, you can write with them today.

People get confused about them because there's a tie-in with the old Germanic religions where they're used by the gods for divination, and the neopagans have adopted them for that purpose. But they're really just a set of alphabets optimized for carving into wood.

> writing is a ... correspondence between marks and elements of human language

Yes, this is what I had in mind by saying "speech", but you're right, the connection to language is the essential part, and sound just happens to be the paradigmatic medium of human language