The biggest difference is Koeman has many promising youngsters worth trusting. Puig included. And he's giving them chances.
Valverde didn't. I'm sure he was well aware of his skill level and opted for better players (even though they weren't great). Gave chance quickly to Fati when he came around. Etc.
Koeman put himself in a position where he has to play youngsters. He let Rakitic and Vidal go, he asked Suarez to leave. Those were Valverde's undisputed starters. No reason to believe they'd leave if Valverde was still the coach, all the reasons in the world to believe that they'd still be here and would still be starters if Valverde was still the coach.
It seems like the initial plan for Pedri was to loan him out this season, but Koeman made him a starter instead. Koeman also brought a 19 years old Dest and shows him tremendous confidence. Koeman is also not afraid to start Mingueza and Araujo instead of Lenglet and Umtiti.
Valverde on the other hand, played old players and his transfer requests were the likes of Willian, Parejo, Murillo and Inigo Martinez.
He played non stop Coutinho on the wing who's horrible there, while he had Malcom and even Dembele when fit. Both were better than Coutinho. Arthur in his second season became a bench warmer, after a first promising year at Barcelona.
No need to rewrite history, Valverde wouldn't trust a youngster to save his life, except when the youngster is a generational talent (Ansu).
Now you're saying that Koeman has promising talents, it doesn't matter. You either believe the youngsters or you don't. Pep played Cuenca and Tello, both turned out to be trash but it doesn't matter, he still gave them a chance to prove themselves.
Maybe in 3 years, Dest, Mingueza, Araujo and Pedri will end up being flops, just like Alena. It's not like they are already proven players. There is still a possibility that they will fail. Koeman gives them confidence regardless of everything. That's how it works with youngsters.
Thing is Rakitic and Vidal were all better than Alena. Of course you would play a better player more.
Alena actually played 27 apps in 2018/19 season. More than Puig btw.
My bad. Alena did play more in that season than Puig currently indeed. But still, Koeman trusts the youngsters a lot lot more than Valverde, it's not even comparable.