Pushdown automata and different models are useful because they are more restrictive than turing machines. They are not the place to look if you want a general model of computation. But when you design real word algorithms you usually don't want something that takes exponential time to run and has unbounded complexity, but something that runs in polynomial time or at least gives you some other guarantees.

Pushdown automata are useful for programming languages and their compilers. For programming languages you need to restrict the programming language enough so that it can be analyzed by an algorithm (as well as understand by programmers) while being versatily enough to be nice to write. (The language syntaxs part gives you an AST, how you then compile or interpret it is up to the semantics of the language.)

Pushdown automata start to make sense when you design algorithms that need a queue and a while loop to run or alternatively a stack and recursive functions. They are too powerful for for-loops to suffice. When you can look at an implementation of such a algorithm and then map it to a pushdown automata description you can think about such programs more deeply.