The very next line says that some things may have additional requirements or exemptions to those rules. If losing access to Game Abilities also makes you lose access to the ability to Focus, then it should be plainly stated that Focus itself is a Game Ability, and that fourth point should be placed elsewhere in the description, because the entire section as written says that loss of Game Abilities doesn't mean you lose the ability to Focus, only that losing Game Abilities might interrupt your Focus, with the implication that this only happens if the ability you're using requires a skill in the first place, like Refitting Armor or First Aid. Magic Items are notable in that they don't require access to Game Abilities to use, barring Magic Items that enhance or alter the effects of skills when used.
To be blunt: Focus, as written, is not a Game Ability, and therefore losing access to Game Abilities has no bearing on your ability to Focus.
In the specific context of this discussion, Arcane Armor specifically says you don't need Blacksmith to Refit it, which means it requires no skills to perform the Refit, and it's a Magic Item, so it can still be used while you don't have access to Game Abilities; put together, that means you can Refit Arcane Armor so long as you are able to Focus, and fulfill the requirement of kneeling while you do it. Losing access to Game Abilities, therefore, doesn't interrupt your Arcane Armor Refit unless you break the kneeling requirement.
Recharge Prowess, in the same vein, only requires that the skill or spell being recovered is on your base character card, and not granted by a Magic Item or other temporary effect. Losing access to Game Abilities doesn't remove them from your card, it just means you can't use or expend them, and Recharge Prowess specifically does neither of those things. Again, there's nothing in the rules around either Recharge Prowess or Focus that indicates that losing access to Game Abilities interrupts or prevents the use of this Magic Item, unless you break the roleplay requirement to regain the skill or spell.
To be fair, there may be effects that cause the loss of Game Abilities whose roleplay requirements would likely preclude the ability to use Focus, such as Enfeeble (Feeblemind) and Prison, but those effects only prevent you from Focusing by prevent you from fulfilling the RP requirements of abilities that rely on it, not by removing your ability to Focus entirely.
To be perfectly clear, I genuinely don't think this is the intended way for Focus to work, but unless there's something I'm missing, this is how it actually works.