It likely shouldn't, but this is an edge case because of the origin of the content
As you've quoted
If a creature assumes the form of a legendary creature, such as through a spell, it doesn’t gain that form’s legendary actions, lair actions, or regional effects.
should still apply to conjure fey because you are conjuring a fey via a spell that then assumes the form of a legendary creature
You summon a fey creature of challenge rating 6 or lower, or a fey spirit that takes the form of a beast of challenge rating 6 or lower.
Fey Creature vs Fey Spirit
I won't speculate on designer intent, but it is likely the options for Fey Creature vs Fey Spirit are functional to separate out your conjuring options:
- Fey Creature lets you pick a creature of the Fey type of CR 6 or lower
- Fey Spirit lets you pick a beast of CR 6 or lower
But functionally, a spirit is not different than a creature. Spirit isn't a creature type, but there are many 'spirit' creatures in the monster manual. Spirit may be a type of Fey, and a Fey is a creature.
A note on conjure spells
While most tables likely let the player pick what is conjured, that is not necessarily the way the designers intended.
A further note on the material
The AL legal supplement cited is not necessarily going to be available at every table. Each DM at each table can decide what content they allow and what they don't. Just because it exists doesn't mean it's an always-on option. That still is up to the DM.
While AL legal, this is not content produced by WoTC, and there are no WoTC Fey creatures that normally qualify that have legendary actions.
Having said that, my very brief look at the supplement does make it seem pretty cool :)