Hello,
that misses the point. Bulletin board systems were not forums, but their precursors. And they did not use global access like the Internet, but point-to-point access via telephone modems. They only had a local network structure, in contrast to the global networked platform of the Internet. But even there, long-term archiving already existed, albeit in a limited form. Perhaps the most important BBS was CBBS in 1978, but it was a forerunner, not an Internet forum in the strict sense.
Conclusion: BBSs were not Internet forums, but they fulfilled many of the social and functional roles that forums later took on on the Internet. So they are ancestors, but not equivalent counterparts.
The Planet Forum was, strictly speaking, a computer conference system or community system. It was network-based, but did not use the Internet, instead using decentralised academic networks.
EIES was a non-public computer conference system that ran on distributed networks and is considered another precursor to Internet forums.
Usenet was conceived in 1979, but did not actually launch until 1980.
Regarding the core question at hand, all of these precursors, like modern Internet forums, also had an archiving function. The conclusion is that there has never been an Internet forum without an archiving function; this has been an important basic function from the very beginning.
Kind regards
CdB