How would a theory of computation course that culminated in lambda-calculus as "the" model of computation, instead of Turing machines, look like?
The kind of approach to theory of computation you describe is what I like to call an abstract machine based computability theory:
i.e. a theory that define computable functions/languages/etc via some abstract kind of machine (automata, linear automata, Turing
machine etc).

An approach that uses $\lambda$-calculus instead of Turing machines could be thought as an expressions based computability theory:
i.e. a theory where one prescribe some basic operations which are intuitively calculable and some operators defined between them to
build other computable operations.

In this expression based computability theory


regular expressions correspond to finite state automata;                                                                             
linear and context free grammar correspond to pushdown automaton and linear bounded automaton
(untyped) $\lambda$-calculus correspond to Turing machines
I don't know very much about complexity in $\lambda$-calculus but I guess  it could be approached as done for complexity of $\mu$-recursive function or of logic formulas, classifying $\lambda$-terms in complexity
classes by their structure (then of course it would be nice to compare these complexity classes with complexity classes for other models of computation, like Turing machines).


Hope this helps.