I'm not sure I agree with your first premise. There are fairly trivial combinators that you can write in lambda calculus that are conditional flow control (i.e. reduce to either the left or right lambda expression based on a value). The implementation of ifTrue in Smalltalk (loosely) follows this model.
More mundanely, the statement is obviously false because a language constructed with the basic arithmetic operators and unconditional branches is also turing complete.
Only if the unconditional branch is a computed branch. Otherwise how would you implement a program that either terminates or does not terminate based on user input? The example that comes to mind is the x86 MOV instruction which, with a single unconditional backwards branch is Turing complete, but this relies on several other aspects of the surroundings that allow you to implement a conditional branch (or, at least, a select, which is morally equivalent).
The simplest Turing-complete instruction set is a subtract-and-branch-if-not-negative instruction, but this is a conditional branch.
I agree that conditional flow control is slightly too broad a requirement, as it depends on an imperative model. Conditional execution depending on input data might be a better way of phrasing it.