You're right that UK citizenship law says you only qualify for citizenship by birth if one of your parents is already a citizen. However, it's more than that when talking about playing for a national team. The UK Home Nations FAs (i.e. all of the different Associations inside the UK) have a written agreement between themselves that they can only use players for national team games who were born British, or who have at the very least a British grandparent. They expressly are forbidden from calling up a player who has himself naturalised and accepted British citizenship at a later point in his life, as the Home Nations view this as essentially cheating and skipping the requirement for national teams to raise their own talent, though they only hold themselves to this rule and not anyone else.
In fact it's more than that - they can only recruit people whose British heritage relates to the area of the UK the team represents. If Gio was born British then he could play for England or Scotland, because his dad lived ("settled", as you say) in both of those countries, but he couldn't represent Wales or Northern Ireland because no Reyna has ever lived there, even though legally citizenship does not distinguish.
I recall a few years ago all this became a particularly big story when England was reportedly considering trying to persuade Mikel Arteta and Manuel Almunia to take British citizenship to play for the national side - Scotland promptly stepped in and threatened to report them to FIFA for breaking the Home Nations agreement, which nixed the whole plan. There also tends to be much tutting and irritation when, say, a Welsh player opts to play for England just because his grandma was English, with the (say) Welsh FA generally believing that the player should consider him Welsh and nothing else.
At the end of the day, let's just say that it's highly unlikely Gio will ever play for a British side, so don't worry about him defecting from the US.
