What helped Surtsey survive storms while nearby Surtseyan islets vanished?

Answer: Lava armored loose ash

Salt glued the cone — Salt spray can cement some coastal surfaces over long times, but it was not the decisive early shield in Surtsey's eruption. Fresh Surtseyan cones are largely loose tephra, which waves can chew away quickly. The stronger protection came when later lava flows covered parts of the cone like a hard cap.

Deeper water blocked waves — Deeper water would not protect a new island from waves once it has breached the surface. In fact, the nearby satellite cones briefly became islands and then were destroyed by wave action within weeks after activity stopped. Their problem was not too much depth; it was too little durable armor.

Lava armored loose ash ✓ — Surtsey alternated wet explosive phases with later lava flows, and those lava flows armored parts of the tephra cone. Nature notes that Syrtlingur and Jolnir emerged briefly but were erased by waves, leaving submarine flat tops. The lesson is that a volcano can win by becoming less explosive and more boring.