Listening!
And, yeah, it’s a bit of an odd duck and no mistake. To answer a few questions so far:
1. The core/peripheral split for characters is intended to be a matter of how often they’re used – the Defuse/Assault/Unleash/Find Common Ground/Lend Aid set are usable in most situations, whereas Forge a Path, Unearth Forgotten Lore and Familiar Face are more intended as useful tools to add spice to play. If you really want to delve into the truth behind a particular piece of weird Tech, you use Unearth Forgotten Lore (and I accept the trigger for that one, in particular, is pretty hinky).
2. The intended dynamic with Family play is that you move to it when you want to advance the overall state of the world, and use Family moves to set up problems or opportunities in the world for the characters to solve.
3. For ‘tell me more’ moves, bad experiences with Discern Realities made me want to try out not giving players one and handling it through the Gumshoe route – if you look, the GM will tell you what you find (see p. 52). These days I’ve cooled on that a little, and realise that they serve as ‘tell me how this situation is going to get interesting‘ levers. As for having multiple of them at the Family level – we wanted a distinction between finding information from someone else (diplomacy) and cashing in Data to come to a realisation yourself and proactively adding things to the map (Uncover Secrets).

Thanks so much for this insightful writeup – I’ve lived Legacy for a half-dozen years now so it’s sometimes hard to see its idiosyncracies. Do tag me into future ones, as it’s always useful to see what jumps out as a stumbling block or an awesome feature to people coming at the game with fresh eyes!