Why models of computation are primarily focused on machines?
I suspect that there are two reasons:


Historical reasons. This is how we've always done it.  Academia can be slow to change.  Also, machines provide a platform for studying concepts like the power of non-determinism.  We don't know how to fully understand the power of non-determinism when considering realistic models of computation (for instance, we haven't resolved the P vs NP question), but when we look at more limited models of computation, it's possible to fully work out the power of nondeterminism.  So, this gives us a platform for understanding some interesting concepts about computation.
Pragmatic reasons. DFAs, NDFAs, and PDAs are useful and important to understand if you want to build a lexer or parser, so they're very useful when building compilers.  A few decades ago, this was a powerful motivation for studying the subject: here is a practical need (how do we structure the front-end of a compiler?) and one where there is some beautiful theory that is directly applicable.  Thus, such a course might gain support both from theoreticians who are most interested in the theory, and from compilers folks who are motivated by practical concerns.  It's not clear that this is still the most important thing for students to learn these days -- today, how many students are ever going to need to write a lexer or parser for a compiler? -- but academic courses are slow to change.  


I'm not trying to argue this is the only reasonable way to do it.  A course on variants of the lambda calculus might be quite interesting, too.  But I suspect these factors might partly explain why such courses are common.