Having to slog through lvl 50 quests is tedious

[Iscah]

The HW story actually starts as early as in 2.3 - when we start looking for "Lady Iceheart"

By that logic, the Heavensward story actually starts right back in the Coerthas story arc (Lv35-40ish), which introduces some important characters and the general 'idea' of Ishgard and its problems.

And then there's the dragoon quests. If you are not yet at Heavensward and intend to play dragoon at some point, this is a good time to do that questline. The Lv30-50 dragoon quests are an early introduction to a character who is involved in the Heavensward MSQ, so if you do it later it will be "out of sequence" as if the Heavensward story hasn't happened yet.