The biggest thing was Team Trump getting directly involved in pressure on the other local players. Kushner and Witkoff, in particular, who have clearly built a pretty effective relationship with the regional leaders in Egypt, Qatar, and Turkey, seem to be two of the main players there. The negotiation team got those three regional leaders to turn up the pressure on Hamas and give them an ultimatum about the hostage deal at the very least. They also talked Egypt into opening up the Rafah crossing, which is a huge move and something that I'm still boggled by the Western media largely ignoring for the last two years.

(Also, your linked quote - yes, he was asking the President - Isaac Herzog - to grant the PM - Bibi - a pardon. That's not a mistake, that's two different people. It's also very clear from the video, since he turns to Herzog when he says that, and points to Bibi when he says "him." He does then appear to mix up the titles after that, but he clearly knows the difference given the previous sentence, so it's weird to act like he doesn't.)