I have only tried with Blaze Spawner, but I am unable to place it. I first tried in overworld and then tried returning it to the_nether.

I checked the online help for the Spawner Chest thinking maybe the "rules" had changed. The wording on the page seemed the same, and that picking up a Spawner and placing it should work the same as in 1.12.2.

The graphic does change from 'box with green dot' to 'box with red dot', when I picked up the Spawner; this made me think it was going to work. but no.

Can someone confirm that it works for them? The ability to move Spawners was a convenient feature.

I have seen this as fixed in 1.16.4.