spellcasting – Does having the ability to cast focus spells count for things that require you to be able to cast spells?

Yes

In the Ranger Class Description Warden Spells are defined as a type of focus spell.

Certain feats grant you warden spells, which are a type of focus spells.

And the rules on Focus Spells define this for non-spellcasters:

You gain the ability to Cast a Spell and use any spellcasting actions necessary to cast your focus spells (see below). However, you don’t qualify for feats and other rules that require you to be a spellcaster.

The prerequisites on archetypes are quite precise in what you need to fulfill them. While some call for a specific class (Runelord) others call for the spellcasting class feature (Hellknight Signifier). There are also archetypes that require spell slots (Geomancer) or a spell repertoire (Wellspring Mage).
And then there is those that require the ability to cast spells from a defined tradition (Soulforger) or to cast Focus Spells (Magic Warrior).

Shadowcaster doesn’t specify any of this. It just requires you to be able to cast a spell. Even a Innate Spell gained from a ancestry would fulfill this requirement, as they allow you to cast spells.

You can cast your innate spells even if you aren’t a member of a spellcasting class.