I wasn’t involved with the Canberra class acquisition; I was in AWD at the time. So my knowledge of it is second hand. However, we went for the Kanimblas to convince the government that there was merit in having ships with large internal and embarked forces capacity, a multi spot helo capability and very good comms and command facilities.
The envisioned missions ranged all the way from commanding and executing (relatively small scale) amphibious and lodgement ops, through HADR type missions to being able to provide A based comms and other facilities either when we.had large numbers of troops deployed for some reason, or even when the PM was at something like the South Pacific Forum. Our intention was to convince them of the utility of such platforms as the Canberras would become after the THSS, a similar sort of ship, was cancelled in the very early 90s; and in that I think we were successful. So I imagine the Canberra ops range would be very very similar, although they are much more capable assets to achieve them.
Included in that mission set was over the beach lodgement and then sustainment of a landed force. That works best if you have a well deck. I’m not sure that the ESBs would be capable of that; and in any case I would not send them in harms way in the same way I believe the Canberras could be risked; they are essentially commercial ships with military capabilities grafted on whereas the Canberras are very definitely built as warships. They could, however, undoubtedly do quite a few of the lower intensity missions the Canberras undertake. The one big advantage of the ESBs comes from their heritage - they are not quite as “in your face” as a large flat deck is going to be, so may well have advantages in some political environments.
So for a Navy like the RAN, the Canberras, as all around performers, are probably the better bet. For the USN, with their much different amphibious capability and (at least before Trump) their international involvement the ESBs, as cheaper to run if slightly less versatile platforms do make sense - provided you don’t have to commit them to a hot war.