What you are referring to are non-Markov processes. That is, process with a memory. These can be useful too, but the reason Markovian processes are preferred

What you are referring to are non-Markov processes. That is, process with a memory. These can be useful too, but the reason Markovian processes are preferred is because of the independence (lack of memory). Independence is a sufficient condition for many theorems that use Markov processes to prove things about average behaviour.