Metal Gear Solid and Konami? Was already one of the top partners with Nintendo and Metal Gear was already on the NES.
Final Fantasy? Squaresoft was one of Nintendo's top partners and Final Fantasy was associated with Nintendo first.
Resident Evil was on Saturn too and Capcom was famous due to its work on the NES and SNES.
Playstation simply took the expanding industry and capitalized on Nintendo/Sega's mistakes by taking the strengths of each side (Nintendo's third party support like Squaresoft, Capcom, and Konami) and Sega (edgy marketing, sports/jock gamer focus, violent edgy games, emphasis on genres like racing/fighters/action games, a Sonic the Hedgehog knock-off) and combining it into one.
I knew like 40 people that owned a PS1, virtually all of them owned a NES before it. Plus with the new kids coming into the industry at that time, they needed a new console to buy too, so you basically had a new wave of kids and the NES/Genesis/SNES crowd needing a console too.
The industry was going to expand no matter what because the NES generation was never going to lose interest in games so quickly. The only "big" series that really brought a lot new to the industry because of PS1 IMO was Gran Turismo. Final Fantasy VII, Resident Evil, MGS, etc. would've just been on Saturn or Nintendo would've been pushed by third parties and finally come to their senses and made a CD-drive for the N64.