I see that, for example:
A cleric or paladin can use a holy symbol as a spellcasting focus, as described in chapter 10. To use the symbol in this way, the caster must hold it in hand, wear it visibly, or bear it on a shield.
Further, I see that the standard arcane focuses — orbs, crystals, rods, etc. — all have to be held in the hand. That makes sense to me.
What I’m not sure about is, for example, the Hat of Wizardry or the Dark Shard Amulet from XGTE. Both specify they can be used as a spellcasting focus (for wizards and warlocks respectively), but it does not make sense to me that e.g. a wizard would be required to hold their Hat of Wizardry in their hand to use it as a spellcasting focus, instead of wearing it. I can imagine a warlock having to grip their Dark Shard Amulet with a free hand, but that still seems like a bit of a stretch, especially when combined with somatic components.
Can these specific foci be used without holding them in hand?