How does the linguistic concept of “time depth” compare to the intuition of “language age”?

Not very well.

Linguists have an understanding of some languages being more conservative in certain aspects than others. Informed by history, they also have a notion of how far back two languages branched apart.

Linguists are quite reluctant to make the further claim that one language is overall more archaic than another, compared to their common source. Often however this is demonstrably true. I have edited a paper, for example, which used state automata to derive Cantonese and Mandarin phonology from Middle Chinese, and computed a quite reasonable metric showing that Cantonese is more archaic than Mandarin.

So much for linguists’ notion of time depth. Lay notions of “old languages” are not derived from meticulous comparison of state automata, of course. Much of the time, they are not derived from linguistic information at all, but from cultural narratives (myths) and ethnic prejudices.

In the best case, laypeople have been exposed to an older language or stage of the same language, through religion or education. If they are paying attention and are multilingual, they can form an impression of which language (or dialect) that they are familiar with sounds more like the archaic language. And their intuitions will be reasonable.

Not infallible though.

In the 1560s, the German scholar Martin Crusius decided he’d learn modern Greek, and corresponded with a large number of scholars in Greece to do so. Theodosios Zygomalas was his main source of information. When Zygomalas outlined the dialects of modern Greek to Crusius, he wrote that the most tragically corrupt dialect of all was Athenian. A lamentable Falling Away from what had been.

The Athenian dialect of Modern Greek was an enclave surrounded by Albanian speakers since the Black Death. Phonetically, it was in fact more archaic than “standard” Greek, particularily in its pronunciation of upsilon. But of course, Zygomalas did not know that. All he knew was, Athenian dialect sounded funny, and historically it was not supposed to. Ergo, it was more corrupt.