I actually think Daniel Jones is in a better situation than Barrett. He is in the Porzingis situation, crucified on draft night, so anything even remotely decent he does on the field will be viewed more brightly since the incoming expectations were so low, and the fans will rally around him, since he's the under dog in a way. This exact situation played out with Porzingis- once the fans realized he wasn't just complete ass they warmed up to him quickly. It happened as quickly as summer league, the fans started going bananas for him because he could dunk in transition and hit threes. And since it is New York, he doesn't have to be great to be perceived as great, because the NY hype train will just go wild and push that talk.
Barrett is entering with very high expectations which will be tougher to live up, making the situation for more delicate than either Porzingis or Jones. However, Barrett made a phenomenal pre-draft decision to declare his love for the Knicks. He made public comments about wanting to be with the Knicks, he only worked out and interviewed with the Knicks, and they're really pumping the narrative of how since he was 12 he wanted to be a Knick, and that his dad played for Saint John's and his mother attended there as well. It's great PR which will buy RJ a lot of time.
The Knicks are complete shit now, yesterday, and tomorrow and the fan base just wants to be wanted after being rejected and tormented by so many for so long. The fact that Barrett wanted the Knicks will buy him so much time from Knick fans.
edit: I still believe Jones is going to surprise a lot of people and turn into a very good NFL QB, possibly as soon as his rookie season. We will see.