Sum of 3 integers with full adder
It seems you have almost invented the carry-save adder.

You can use independent full adders for several layer, each time reducing 3 wires of equal weight to 1 wire of that weight and 1 for the next higher weight. Of course if there are 6 wires of the same weight, you can reduce them to 2 of the same weight and 2 of the next higher weight, and so on.

But the result of doing a bunch of that is not the sum, but two partial sums, one coming from all the S outputs of the last layer and one coming from all the C outputs. They still have to be added normally. You can use a ripple-carry adder or something faster/bigger there like a Kogge-Stone adder (which you'd only need once, compared to n-1 times if you just added up everything the obvious way).