It's an attractive thought, this comparative look at FA and TM. However it is misleading to an extent.

A Turing Machine's capabilities exceed the FA's because it has a 'store of knowledge'. It is both recording and reading, repeatedly. There is erasure, too. So if we're aspiring to TM-hood, the guidance seems to be write, and re-read (your own thoughts), and re-evaluate, and re-write. This process is subject to the halting problem. /g