[consilient]

> Does the central place that Turing machines and the idea of a linear tape retain in CS reflect some deep mathematical truth

No, not at all.

> or is it a cultural preference based on Turing's key historical role?

A little bit, but the main reason is just that Turing machines are a convenient language for introducing complexity theory, which is usually the first and often the only area of CS theory encountered by undergrads. Other areas are fit for other tools. Computability theory typically works with mu-recursive functions, programming language theory with lambda calculi, computer architecture with various sorts of register machines - Turing machines have a central place in CS but not the central place, by any means.