For those who would like to avoid Android completely, there are some alternatives in the form of
Linux smartphones.
Linux Smartphones are worth to be mentioned in the space of morden world privacy phones, the are a smart alternative too. But one cannot use android as suggested in the OP mostly, as most of the people use it as a daily driver and google provides one of the best data storing techniques and best user experiences in there daily driving apps.
Btw, even if you use custom roms, using Google apps is still important.