Is conditional branching a requirement of Turing-completeness?

Question

I\'ve been searching the web and I\'m finding somewhat contradictory answers. Some sources assert that a language/machine/what-have-you is Turing complete if and only if it has

Answer 1

You don't need conditional branching to build a Turing-complete machine, but of course any Turing-complete machine will provide conditional branching as a core feature.

It was proved that systems as simple as the Rule 110 Cellular Automaton can be used to implement a Turing machine. You sure don't need conditional branching to pull such a system from the bit bucket. Actually one could just use a bunch of rocks.

The point is that a Turing machine will provide the conditional branching, so what you are doing anyway by proving Turing completeness is somewhat implementing conditional branching. You have to do it without conditional branching at some point, be it rocks or PN-junctions in semi-conductors.