How do I get adjustment layers in After Effects apply to video clips in the Premiere timeline?
That's simply not possible. Your AE comp is merely a "container" that doesn't respond to anything that happens natively in Premiere and only returns a pixel buffer (frame/ image). Any such stuff will have to be done directly in AE.
What a bummer. Is this really not possible at all? It would make animated graphics with a blurred background much easier to implement than a separate animated adjustment layer within Premiere. Would much rather just animate an adjustment layer within the AE comp, that syncs with the animated graphic within AE.