In Italian, figuratively speaking, a "caryatid" is used to define an "immobile" person fixed on his outdated ideas and beliefs.
Thanks for explaining your intended reference. My Oxford English reference was "a stone carving of a draped female figure, used as a pillar to support the entablature of a Greek or Greek-style building." The symbolism is somewhat different, but I can see how we get to your figurative usage from there.
A key thematic difference of course is that architectural caryatids are female. Our immobile ASR codgers not so much. Tangentially, immobility (and otherwise) was a key element of the deadly statues in that Doctor Who sequence.