[nonsequitur]

This doesn't work because you're not left-shifting (doubling) the carry. But when adding the shifted carry to (x ^ y) we're back to potentially overflowing the highest bits. The solution is to add the highest and the lower bits separately:

  lower = 0x7f7f7f7f;
  highest = ~lower;
  z = ((x & lower) + (y & lower)) ^ ((x ^ y) & highest);

Note this only improves performance for larger container integers.

[rustybolt]

You're completely right! Typically you don't care about overflow though (and you should use unsigned ints to avoid undefined behavior).