The most mainstream games that featured Forge philosophy were D&D4 and WFRP3, both of which were failures compared to previous editions.
There are a lot of new-school games that feature narrative mechanics, however, the "Father of Narrative RPGs" isn't Ron Edwards, but Greg Stafford and Robin Laws. The most current successful narrative game, Numenera, is much more Laws then Edwards.
The Forge was a radical factionalized section of gaming theory based on a misunderstanding of the Threefold Model that was as much a Cult of Personality as it was a design movement.
Did it have an impact? Yes, but that impact was a failure. Even Baker designed his Xworld system to get away from the mechanics and designs that came out of the Forge.
However, Narrative gaming, or I should say, "Roleplaying Games" that feature a large amount of OOC mechanics whether for narrative or tactical reasons is very much a hallmark of the new school and it is troubling I think that there are no successful new games that are not new school, Next being an exception if it sells well, but still relying on the D&D name.
There is no traditional Numenera, for example.