Recording ~5 hours of television a night would have been a trivial cost for a network like NBC. Particularly compared to the licensing fees those hours would have had.
In general, the affiliate nature would add a wrinkle to the whole thing, but not an insurmountable one. If nothing else, they could have used the broadcast from the affiliates they owned.
Broadcast-grade video tape cassettes were expensive even in bulk, and all tape requires climate-controlled archival storage since heat and especially humidity are quickly destructive to the adhesives that hold the magnetic layer to the substrate. (If you'd like more detail here, the term of art for this failure mode is "sticky-shed syndrome". While it's obviously more of a problem now than then, archival needs were understood at the time.)
Depending on format (Betacam SP or U-Matic), an open-ended commitment to preserving all programming would involve adding at minimum 2/3 to 1 cassette per hour of programming - more if you want multiple copies. So your running costs start out sizable and only grow over time, in search of a highly speculative payoff that at best won't be realized for years to decades.