Adders belong to category of combinational logic, which by definition excludes memory. So OP is right, they do not yet demonstrate Turing Completeness. I imagine, but don't actually know, that they would be about as expressive as dfa, which are few steps below turing machines
He builds the adders from ANDs, ORs and NOTs. With these, you can also build flip-flops, which are memory.
(Actually, it's enough to have only NAND (or NOR) to build all logic circuits, including memory)
[0] https://en.wikibooks.org/wiki/Electronics/Flip_Flops