On the one hand I think the current situation should be the default so it's compatible with existing code and practises.
There is no requirement in this thread to come up with something that would be fully compatible with C. Otherwise, the better approach would probably be to try and get involved with the C std committee, and make proposals. The probability of being able to really make a change is pretty low though. C as a language has to keep the most complete backward compatibility possible. It's one of its strengths, but it does of course prevent a number of features to be added in order not to break existing code.
So the keyword would be 'checked' rather than 'unchecked'. But, as noted above, it requires a bit of discipline and would be left out by default more than it would be put in. So on the other hand having to explicitly decide it's unchecked would be better 
The whole point of this kind of feature is to make it default. It's like making objects immutable by default in Rust. If the "more secure" approach is not default, then it's basically useless.
Error handling is related to this in that regard. I think there are definitely two categories of programming approaches out there.
The first one is to design for success. In this approach, it's considered that the normal execution flow is the one which is perfectly successful with no error or failure whatsoever. This has been coined the (in)famous "happy path". This is also the main idea behind traditional exceptions as a way of handling errors. Failure is considered exceptional; success it the normal path and is what should be favored.
The second one is to design for failure. Here, it's considered that failure will happen and that it's an integral part of "normal" execution. In this approach, you do careful error handling, you usually exclude exceptions (failure is definitely not considered an "exception"), and overall, error handling and input/parameter validation are an integral part of the code. They are considered every bit as important as the "happy path", and thus there is no reason to favor the latter and to try and "hide" error handling.
I'm more for the second one, and I wish the "happy path" concept would die a painful death.
