Why transcripts instead of degree certificates? I've been offered a few adjunct teaching positions recently, and the colleges have said that degree certificates don't count in verifying my credentials and that only academic transcripts are acceptable.

Given that I finished my degrees 20+ years ago, what is the reason for this?

I eventually managed to get my transcripts, but no one has ever asked for them before, my degree certificates were perfectly acceptable (including tenure-track positions in Australia).
Most applicants for positions at US universities will have been educated in the US, so US hiring practices are based around the sorts of records that US universities provide.  And US universities don't really have such a thing as "degree certificates".   The paper diploma from a US university is considered purely ceremonial and not used for any official purpose; it often doesn't even have complete information about the degree (major, honors, etc).  For most US universities, the only official academic record they produce is the transcript.  Hence, that is what a US university expects when hiring.

Note that in some cases, a hiring decision may be based on more specific details about your education than the simple fact that you have a degree.  There may be formal requirements, coming from university regulations or accrediting agencies, that you have a certain amount of coursework in certain areas, and the transcript is the only way to verify that.