Caveat: below I am only saying that it could be done, not that it should. I am in the nets/sequences first camp.

I think implicit in the sentiment behind this comment is

How can you define convergence of a series without referring to the convergence of its sequence of partial sums.

Because as we know, sequences are telescoping series are practically indistinguishable.

And I think in principle this could be done, and in some way can be a good motivation for the notion of completeness.

Essentially, one can start by motivating the idea that a series converges (settles down to a number) if its tail becomes "negligible", meaning that all the changes past a certain point is very small. This one can expand to meaning "past a certain point, the sum of any consecutive string of numbers in the series is no more than a fixed error". (Basically start by motivating, the definition for the series being Cauchy.)

Then one can start asking the question of whether there actually is a number that is being represented by this infinite sum (Cauchy completeness).

BTW, thinking of a sequence as a telescoping series is occasionally useful in analysis (most textbook proofs of Banach fixed point, for example).