FFT based symbol synchronization for digital demodulation
Symbol timing synchronization seems to be a complex topic although once you get some basic principles right, it all makes simple sense. The method you have referred to is known as Digital Filter and Square Timing Recovery$\ ^{[1]}$, also referred to as Oerder and Meyr algorithm.
EDIT:
And later steps in your summary are not correct. There is no search, timing phase is extracted from the first harmonic. Everything else except the DC term is zero due to the signal being bandlimited.

[1]: M. Oerder and H. Meyr, &quot;Digital filter and square timing recovery,&quot; in IEEE Transactions on Communications, vol. 36, no. 5, pp. 605-612, May 1988.