What definition takes up fewer components in a digital circuit is a terrible reason. The whole point of math is we can reason about the most conceptually simple idea, rather than with engineering constraints. Sets existed before circuits! And before digital the only “hardware representation” was an analog voltage, which cannot easily represent a pair.

Also it’s not even true. There is no hardware representation for the ordered pair containing the earth and the moon. You now need a bit encoding of the information.

The distinctions of infinite constructions you mention are already well understood. See “recursively enumerable set”.

Ordered pairs are trivially definable in terms of sets. It’s a distinction which does not change any of the foundational proofs and gives you no new insight. This is like arguing that bounded vs counted ranges are foundationally important. We can show they are equivalent in one paragraph and move on.

An actually new ideas will give new results.