This is partial as I am sure someone else will come along with a project that is closer to the one OP is asking about:
The RemoveDEBRIS project
..not quite the same but the closest i have seen is the 'net' test carried out a few years ago:
An untethered test, the design would actually be tethered to a mother craft, to tow, drag, de-orbit, etc.
Net expands to ensnare target object (footage filmed from parent satellite)
Net captures target object (top right of net)
Net is wrapped around and constricts the target object.
The target was initially spinning and you can see the momentum is transferred.
Mission simulation:
In the simulation you can see that it sort of matches the OP's suggestion in that:
..the net is initially collapsed inside a container. As it is ejected from the container, a mechanism opens the wires into a cage (or web) much larger than the object to be grabbed. Once the object is positioned inside the cage, the momentum will go some way to dragging the web or net around the object, but in effect capturing it.
as seen in the real test, there is still a 'sudden transfer of angular momentum between debris and...' the net or web, but with it being tethered you could conceive that the same momentum might be dissipated in some way before it gets to the mother spacecraft.
Website:
https://www.surrey.ac.uk/surrey-space-centre/missions/removedebris
Description:
Net experiment and cubesat
The experiment is designed to help mature net capture technology in space. In this experiment, the first CubeSat (produced by Surrey Space Centre), DS-1, is ejected by the platform at a low velocity. DS-1 proceeds to inflate a balloon which, as well as acting as a deorbiting technology, provides a larger target area.
A net (produced by Airbus Defence and Space) is then ejected from the platform when DS-1 is at 7m distance. Once the net hits the target, deployment masses at the end of the net wrap and entangles the target and motor driven winches reel in the neck of the net preventing re-opening of the net.
From notes:
spinning is caused by a cold gas generator onboard the target cubesat as a result of the inflatable.
For future missions the NET would be tethered to the spacecraft.
For this mission we are in a relatively low orbit and the extra drag from the NET will pull the debris down fairly quickly anyway.
Article:
https://www.thinkdefence.co.uk/2021/04/defence-applications-of-space-debris-removal-systems/
The UK, with its very strong presence in the satellite sector, has the Remove Debris project led by the Surrey Space Centre