I love Iterative Design, does this get banished by my Jhin legend or banished by Flow?

Ruling: It's banished by Flow, not by Virtuoso — so it does not count toward Jhin's four banished spells.

Flow's "Then banish it" is a delayed replacement effect that sends the spell to Banishment as it leaves the Chain after resolving — that happens before Virtuoso's "when you play a spell" trigger even gets to act (Rule 829.1.b.1; same principle as Time Warp, FAQ #8769). On top of that, you only paid 2 energy via the Flow cost [2][C], so Virtuoso's "if you spent [4] or more" check fails anyway (see the Applied Researcher ruling, FAQ #11765).

The card still counts as a spell you played (so "play a spell" triggers fire), but its banishment is Flow's, not Jhin's.