No --- purgatory is man's sinful refusal to accept the death of Jesus on the cross as full payment for their sins --- all of their sins.
the people in Purgatory ... are awaiting the day that they are totally pure and can enter heaven.
If that's what they are waiting for, then they will be waiting an eternity for a day that will never come.
Likewise, the people in Purgatory know they are NOT going to hell!
Right --- because they are already there.
Well, for the sake of completeness, a critical aspect of the definition of Hell is that there's no getting out. Purgatory, by contrast, is NOT for ever. It's until you're cured or paid up or whatever metaphor one likes. So everyone in Purgatory is assured that they are going to Heaven. By definition, therefore, they are not already in Hell.
Side note: calling something "overwhelming" does not make it so.