Is there an acceptance that there may be more incredible finds buried beneath modern Rome and London, for example, that are now lost to us due to large-scale urban development?
Alternatively, as the rate of these finds slowed over the course of the 20th and early 21st centuries, is there a feeling that we’ve uncovered most of what was left behind for us to find?
There is absolutely, positively, without a doubt a vast array of ancient remains buried beneath many cities of the world. You mentioned Rome: right now the construction of a new metro line adjacent to the forum is revealing many new finds. Digging anywhere in Rome will produce ancient remains, just about.
The trouble with modern cities which overlay ancient sites is that they are full of people, which obviously limits access. Modern excavation is slow, meticulous, and expensive, and once archaeological material is revealed, it must be conserved and protected. This is also very expensive. In fact, development in cities which overlay archaeological sites is generally onerous and expensive, and often crews will neglect to report what they have found because it will cost the company so much money. The circus complex at Carthage and an adjoining cemetery were nearly bulldozed because the Tunisian authorities just wanted to get their road built as quickly and as cheaply as possible. This guy came to light because of the subsequent rescue excavations.
Another factor is: what are you looking for? There might be an undiscovered Roman theater in your city, but the excavation costs associated with uncovering a massive Roman theater in the middle of a city are not generally worth it, unless you have a specific research goal in mind. We have already excavated many Roman theaters.
There are many places where modern buildings are "in the way" of major, important discoveries, however. Example: Cadiz, in Spain, was a major Phoenician city already in the 7th century BCE, and we know virtually nothing about it. Any sort of excavation at all in Cadiz which can bring to light Phoenician material from the early Iron Age would be extremely interesting. Likewise, more excavation at Pithekoussai: the Greeks and Phoenicians had a "joint" settlement experiment here on the isle of Ischia, Bay of Naples, in the late 8th century BCE. It was abandoned suddenly, probably due to volcanic/earthquake activity. We have extensive excavations from the necropolis and a "trash dump," but none from the settlement itself, which is now under a massive spa resort. I would love to bulldoze that resort entirely and drop some exploratory trenches.