You are asking several different questions. Let me briefly answer them one by one.
What is so important about the Turing machine model?
During the infancy of computability theory, several models of computation were suggested, in various contexts. For example, Gödel, who was trying to understand to which proof systems his incompleteness theorem applies, came up with the formalism of general recursive functions, and Church came up with the $\lambda$ calculus as an attempt at paradox-free foundations for mathematics. Turing himself was motivated by a problem of Hilbert, who asked for a "purely mechanical process" for determining the truth value of a given mathematical statement.
At the time, Turing's attempt at defining computability seemed as the most satisfactory. It eventually turned out that all models of computation described above are equivalent – they all describe the same notion of computability. For historical reasons, Turing's model came out as the most canonical way of defining computability. The model is also very rudimentary and so easy to work with, compared to many other models including the ones listed above.
The usual computer science teaches Turing machines as the definition of computability, and then uses them also to explore complexity theory. But algorithms are analyzed with respect to a more realistic model known as the RAM machine, although this issue is usually swept under the carpet as a secret for the cognoscenti.
Aren't DFAs a better model?
This was the original motivation behind Rabin and Scott's famous paper, Finite automata and their decision problems:
Turing machines are widely considered to be the abstract
prototype of digital computers; workers in the field, however,
have felt more and more that the notion of a Turing
machine is too general to serve as an accurate model of
actual computers. It is well known that even for simple
calculations it is impossible to give an a priori upper
bound on the amount of tape a Turing machine will need
for any given computation. It is precisely this feature that
renders Turing's concept unrealistic.
In the last few years the idea of a finite automaton has
appeared in the literature. These are machines having
only a finite number of internal states that can be used
for memory and computation. The restriction of finiteness
appears to give a better approximation to the idea of
a physical machine. Of course, such machines cannot do
as much as Turing machines, but the advantage of being
able to compute an arbitrary general recursive function
is questionable, since very few of these functions come
up in practical applications.
It turned out, however, that whereas Turing machines are too strong, DFAs are too weak. Nowadays theoreticians prefer the notion of polynomial time computation, though this notion is also not without its problems. That said, DFAs and NFAs still have their uses, chiefly in compilers (used for lexical analysis) and network devices (used for extremely efficient filtering).
Isn't the Turing machine model too limited?
The Church–Turing thesis states that Turing machines capture the physical notion of computability. Yuri Gurevich has led an attempt at proving this thesis, by formulating a more general class of computation devices known as abstract state machines and proving that they are equivalent in power to Turing machines. Perhaps these machines are analogous to your idealized model.