to me, it is a purely a question of semantics.
IF you do not have an alternate, and it was NOT required at the flight planning stage, then you do not have an alternate!
IF.... the wx falls below alternate minima, but still above landing minima, then it is OK to continue.(REF: have a look at ther airport page on Jepps... it says "Alternate, for filing")
IF the wx is now below lnading minima, AND you have no fuel on board to divert, then you don't have too many options.. do you?
IF the wx is below the landing minima and you DO have fuel to divert somewhere, then.... it is up to you!
As to why aircraft end up all over the east coast when the wx turns to custard... well, that is because they DID divert because the airport was unsuitable... not just because the wx guesstimate told them to go and get moer fuel... just in case.
Of course, the other option is that is the hazard report is only for inter/tempo, and you DO have fuel to HOLD, then you do not need an alternate.