WARNING!
This turned out to be way long , detailed, and complicated, but it's raining an I got nothin better to do, except things I SHOULD do. Every time I thought of something I thought of some thing else. Sorry.
Bigger and more downspouts should help especially if you have the too common 2X3 spouts, the big boxes have 3X4, larger sizes from gutter specialists. You can also use thin wall PVC pipe. Big boxes usually don't have any thing but 5" gutter, which is actually about 3 across the bottom and up the back with the front sloping out in that ogee shape (called K style) to about 4" across the top. Larger sizes from folks who do commercial work. Usually larger down spout is more effective than larger gutter except on long gutter runs.
.You do need a spout in that corner, due to the geometry, an inside corner often winds up being too low. And it looks like it is at the bottom of a valley. A lot of water comes rushing down a valley and can over shoot the gutter. You may need a diverter extending up on the outside of the gutter, but they trap debris behind them you can also put a diverter(s) higher up the valley to send the water away from the corner. I am getting way to complicated here, you pro'lly don't need to go to such extremes, cept mebbee a small diverter at the corner.
A couple more spouts wouldn't hurt.One on the garage corner between your 1 and 2, My rule of thumb has been to put a spout at each outside corner of a roof, another one opposite A, that is, 'down' on your great drawing, at that small outside corner. The less water going to A the better.Don't know where you'd drain that one to, p'rhaps a run of spout angling down across the wall to that very bottom spout, UGLY. BTW where does 2 drain to? If you're not getting flooding from 2, A won't have to drain much farther than that.
Now your “French drain,” as CM said it's not really a French drain, which is the perferated type used to collect and distribute ground water, we're talking about surface water. The patio could use some more slope away from walls and corners, but we got what we got. You got all kinds of what look like floor drains and capped off pipes are those part of the swimming pool system, where do they drain to, back to the pool and the sanitary sewer? If so we can't use them to solve the issue. Is there a slope toward the rocks? What is that black circle in the 'rocks'? It sure is tempting if its a drain. Could it be disconnected from the pool, the top lowered, and rerouted? The whole rock area looks like a good collection point for drainage if the patio just slopes a bit toward it. At any rate get the rocks and gravel out, dig and grade the bottom to slope towards the side walk at 2.
( HOKEY SMOKEY!. I just saw in pic two electrical(?) wires. Ground wire, phone, cable? What ever that's not good in the water.. Sorry for the interruption, but...)
Any way grade the bottom as far below the bottom of the side walk as the outside diameter of your SOLID thin wall drain pipe. If we can use it as a collection point make sure there are no jaggedy rocks, roots, broken Scotch bottles, knives nor fossils in there. Cause we gonna line the inside with some really thick plastic sheeting, one big piece is best, pond liner is better but costlier, I once used an old water bed, not the liner, the water bag. Got one of those around? If no collection just dig a trench deep enough for your pipe from the spout.to under the walk.Figure out where 2 is gonna connect. Tunnel under the walk, trench from there to your discharge point.You want a slope of at least 1/8 “ per linear foot, 1/4” is better, more is even better, you want the top of your discharge to be below the bottom of your collector, if just a spout drain below the bottom of spout. Slide your pipe under the walk, a piece a couple of feet longer than the width of walk is easiest. Connect pipes on each end, to spout and discharge, trenches , pack soil and gravel in tunnel, refill the rocks.
I best stop. I was gonna get into detailed instructions for tunneling lining the collection box filling with the right gravel, warping time space continuum and who knows what. Sorry about that. Mebbee you could just toss a humongous sponge out there.
I DO wanta compliment you on the pics and especially the drawing, some verbal discriptions of problems are more confusing than my answers.