> I'm confused about your quote though: is "remarkably minor" meant to mean "less important than expected"?

Yes.

> Were there other forms of "phonetic writing" at play?

At the time that quote was talking about? No, since at that time, proto-cuneiform (as they call it) would have been the only writing system in existence. Go forward a bit, though, and several forms of “phonetic writing” begin to appear frequently: rebuses (as in the quote), syllabic characters (e.g. [baʼugeš] was written BA-UG₇-GE, Edzard 2003), and characters for individual consonants (unattested in Sumerian, but Egyptian used them extensively).

> If yes, then does that mean that there actually were case of symbols appearing directly as written translation of phonetic "concepts"?

Exactly. (Even my earlier quote has examples of that.)