While i don't know the answer for sure, from what I read it seems like they did realize and understand, in the end, that they were not fit to carry The Mantle. They already knew humanity was the chosen race, given that's the reason they rebelled against the Precursors to begin with, but seeing their mistakes they likely felt their time was over and it was humanity's turn.
I always had a theory. It can't be backed up by anything, other than the book "First Strike" where they find the Forerunner Crystal that manipulates slipstream space.
My theory is that they figured out a way to live in slipstream space. Cities, homes, etc. The didn't need to repopulate since they survived the blast by remaining in there. I always thought this since 2004. Well before Halo 4 and Halo Legends. Not saying it is right, but it works for me.
Well according to the first two book of the Forerunner trilogy, the Halo rings firing didn't actually wipe out all of the Forerunners, that wasn't the end of their race. It just destroyed a huge chunk of their capital world but they continued to fight the Flood for years after that. It's implied that the Flood just took over everything Forerunner, including their own seeding Arks and that's how they eventually died out.
By this level in their evolution the Forerunners depended on their armour and ancillas so much that when the Metarchy fell after Mendicant Bias attacked the capital with the Halos, their armour and literally all of their technology began a slow crumble into disrepair.