Both laptops started off with the bluez and bluez-firmware packages plus their dependencies installed as part of the default trixie load.
To get the desired functionality, these three additional packages need to be installed:
bluemanpulseaudio-module-bluetoothlibspa-0.2-bluetooth
Then reboot the system.
Here endeth the fix, some notes follow.
Many internet posts talk about the first two packages in relation to this issue. Only a few mention the third package - this is the one that put me on to it. But none that I found concisely list all three sans various configuration file edits and other things that I did not end up having to do.
There is a package in trixie called bluetooth but it does not install any of the three required packages, so it was useless for my purposes.
This was tested with various cheap bluetooth headsets and they were able to connect, and then both receive and transmit audio.