By 1990 we were finished (the band was no more)
The verb is "were". The past perfect would be "had been". So, you are proposing "By 1990 we had been finished (the band had been no more)." This really doesn't sound right.
You could say "By 1990 we had been broken up for 2 years." That is something (being broken up for a specific time period) which was completed before a point in time.
On the other hand, "being finished" is an ongoing state of being, which by 1990 was still occurring, and it wasn't completed. Therefore, it somehow doesn't match with the past perfect tense.
Something else to observe, is the difference between an active verb versus a participle form:
By 1990 we had finished.
OK. This is past perfect, and an active verb.
By 1990 we had been finished.
No. This is past perfect (of "to be"), and a participle.
So, to change your original sentence into past perfect it should be shifted not only in tense, but also in terms of the above mentioned active/not-active construction.
However, the meaning also changes. "to finish" and "to be finished" are different.
So, a better way to phrase this in the "past perfect" would be:
By 1990 we had broken up.
You are correct, that past perfect goes with "by the time". Generally, yes.
However, what this current discussion seems to be showing is that's not always the case with the verb "to be".