I have a small nit about #2 that doesn't affect your main argument. Similar to the 'overflow' flag the processor automatically sets to signal signed overflow, the processor flag for 'carry' is already being set by the processor when you add or subtract unsigned numbers that overflow. There is no separate signed or unsigned addition in x86/x64 assembly --- the processor just sets the flags for both cases, and they can either be used or ignored.
This is the reason that overflow checking can be so inexpensive --- no additional checking is necessary, just a conditional branch on an existing flag. Correctly predicted branches not taken are very inexpensive, and this branch will almost always be correctly predicted. The issue is that standard C doesn't allow any direct defined way to make use of the overflow flag. Instead you have to write something awkward and hope the compiler optimizes it down for you.
This is the reason that overflow checking can be so inexpensive --- no additional checking is necessary, just a conditional branch on an existing flag. Correctly predicted branches not taken are very inexpensive, and this branch will almost always be correctly predicted. The issue is that standard C doesn't allow any direct defined way to make use of the overflow flag. Instead you have to write something awkward and hope the compiler optimizes it down for you.