WetGeek Today it occurred to me to try using Nemo instead, so I installed it and damned if it didn't work!
I don't know whether this matters to anyone else, but since I made this statement, I thought I should tell why I removed Nemo and replaced Dolphin. Then stopped using that laptop, except as a backup device.
Nemo did indeed give me access to remote folders, allowing me to enter a password to gain that access. But the file manager is also pressed into service by other application. E.g., choosing a destination for exporting an address book from Thunderbird, or accessing the Downloads folder to receive a downloaded .ISO file. For such purposes, it doesn't prompt for password access and thus doesn't work at all.
Since VirtualBox 7.x is to be in next week's sync, I'll just keep waiting, and put up with 6.1 until then. I just didn't want to give the impression that Nemo was the solution to all my file manager problems, as I first thought.