There is no such thing as resistance/immunity to non-magical weapons anymore
The MM errata changed this:
Damage Resistances/Immunities. Throughout the book, instances of “nonmagical weapons” in Damage Resistances/Immunities entries have been replaced with “nonmagical attacks.”
This includes your example of a werewolf which says:
Damage Immunities: Bludgeoning, Piercing, and Slashing from Nonmagical Attacks that aren't Silvered.
So, going forward in this answer, I will ascertain how swift quiver applies to this new wording.
Swift quiver alone doesn't bypass resistance to nonmagical attacks
The reason is simple: swift quiver does not deliver a magical attack.
The same MM errata explainings what magical attacks actually are:
Particular creatures are even resistant or immune to damage from nonmagical attacks (a magical attack is an attack delivered by a spell, a magic item, or another magical source). (MM, p. 8 post errata)
Swift quiver says:
You transmute your quiver so it produces an endless supply of
This explicitly says that the ammo is nonmagical and thus won't count as a magical source for the attack.
You might point to this as a counterpoint:
On each of your turns until the spell ends, you can use a bonus action to make two attacks with a weapon that uses ammunition from the quiver.
However, this is an attack granted by a magical source (a spell), but it is not actually delivered by the spell — it is delivered by your bow. If the bow is nonmagical then it won't count as a magical attack.
Crawford's ruling doesn't apply here
Regardless of damage type, the direct damage of a spell is magical.
Swift quiver does two things: give you infinite nonmagical ammo, and gives you a bonus action you can make extra attacks with. Neither effect does direct damage (nowhere in the spell does it say it does XdY damage) so there is no contradiction in rulings here.
However, arrows would bypass resistance if fired from a magical weapon
Ammunition fired from a magical weapon is considered magical:
If a magic weapon has the ammunition property, ammunition fired from
it is considered magical for the purpose of overcoming resistance and
immunity to non-magical attacks and damage. (DMG 140)
Thus, if you fire any arrows from swift quiver with a magical weapon, those arrows are considered magical and the attack is considered to be delivered by a magical source. Thus, with a magic weapon, the swift quiver would allow you to bypass resistances/immunities to non-magical attacks.