Kind of depends on what we mean when we use the word "Internet". If we mean internet as a social/economic revolution, then sure, BBSs were relevant. If we mean Internet as a set of protocols, software that implements them, and hardware to run them on, then I don't see that BBSs had a lot of relevance (though they used sliding window protocols for file transfer such as ZMODEM/YMODEM).
There was a brief period where BBSs also offered various gateway services into the Internet a long with the normal BBS services like doors. I received my first ever email address from a huge local BBS back in the early early 90s by paying a token fee every month. I had nobody I knew with email addresses until a few years later so it was kind of pointless, but it did work.
On a larger scale, when AOL started offering internet services through their network is sort of the same thing until they eventually just become an ISP.
Big networks were gated in and out of the Internet, such as UUCP and BITNET. (UUCP for Unix systems, BITNET for IBM mainframes.) This had interesting effects on email addresses:
Is that email address even real? I can understand some of it but not all. Looks like UUCP to research, UUCP to ucbvax, somehow send to cmu-cs-pt.arpa, idk what's up with CMU-ITC-LINUS, email to dave%CMU-ITC-LINUS@CMU-CS-PT, which forwards to dave@CMU-ITC-LINUS
That's really interesting! Just to be clear in my post I'm referring to bog standard email addresses not fidonet...which was also very common at the time.
I'm not aware of any fidonet<--> internet bridges during that time period but I suppose their could have been. AFAIK fido relays all occurred via POTS sync calls between BBSs
