Others here are much more experienced as I haven't even been there yet, but having done a fair bit of research here are my thoughts to some of your questions.

Technically, Sorrento is not part of the "Amalfi Coast" as that starts at Positano and goes up to Vietri sul Mare. I know several people who did Pompeii and all from Naples as they went by train and apparently that is easier. If you plan to have a car it may be different.

As far as "chain" or " chain affiliated" hotel options go, there aren't many in the Sorrento and the Amalfi Coast areas. Naples has a fair number and then for access to the Amalfi Coast part there is Salerno (has a Novotel, a Holiday Inn) which is not part of Amalfi coast per se, but is a slightly bigger town nearby Vietri sul Mare which may not be as glamorous, but can provide much more reasonable hotel prices and decent access (also via ferry) to Positano, Amalfi, Ravello, etc. as well as train from / to Naples and Pompeii from what I understand. Obviously no luxe options in Salerno IIRC.

For Hilton there is only the Hilton Sorrento and depending on time of your visit, prices can be pretty high. As I shared in above posts, I'm planning to go early May 2024 and rates at Hilton are already quite high for that hotel starting in the 500 - 1,000++ Euro range depending on room / suite type for what is early in the season still. Global Hotel Alliance (GHA) has the Anantara Amalfi in Amalfi, but for now I see 18 sqm entry level rooms for like 1k Euro which is just nuts though maybe rates will moderate over time.

Via Hyatt you have the SLH hotels (only Angiolieri in Seiano and Cocumella in Sorrento) and depending on the integration also a few Mrs. & Mrs. Smith ones (8 or so) though they can be even more expensive than the SLH hotels easily going into the 1-2K Euro a night range and when we had them under IHG (previously affiliation) rooms for points where very scarce. Maybe with Hyatt there will be better points nights availability, not sure and to be seen.

If you really want to be near Positano, you will have to look outside of bigger chains or chain affiliated hotels as the only one I'm aware of there is Casa Angelina (in Praiano) under Mr. & Mrs. Smith (not yet Hyatt integrated) and in May next year (for example) that one runs ~1.2K Euro per night for the lowest category Romantic Sea View room (23 sqm). YMMV.