Is there any sane reason to actually implement it?
Bare metal code is never sane, but that's not an excuse not to write
it. C is a very complete language, especially modern variations like
C11, but there's still a lot that can't be done in C and for good
reasons. Kernel code and optimal libraries can stretch the compiler a
lot more than user code, and often enough, there's simply no way to
represent ideas in C. One of these examples is unwind code.
That's all fine, but I'm not sure how this supports having named register builtins. The problem is that once we implement a feature, it may be impossible to get rid of it, should it turn out to be a flop. Do we understand all intended and unintended consequences of implementing this?
Then you ask...
Are there any cases when inline asm would work well enough?
Sorry, that was meant to be "would not work well enough".
Well, assembly is very powerful and so, but it's also very hard to
understand what's going on. Inline assembly is an extension to the
language, and because of that, different compilers implement them in
different ways.
True, but I know of only one other compiler that allows register variables (and it's not even a C compiler). The portability argument is not a very strong one here, at least as I understand it.
Most of GCC's implementation is hidden in layers upon
layers of legacy code that not many people dare to touch, but that
compiles code that can't stop being compiled, nor easily migrated (for
technical and legal reasons).
The interaction between inline assembly and C is, therefore, not easy
to implement, and it's even harder to get it "right" (ie. similar to
the "other" compiler). The most you could write in C the better for
*compilers*, and minimising the exposure by using register variables
is actually not a bad idea.
I'm not sure if I'm following your argument. The code that uses inline asm that cannot be easily migrated will likely not be rewritten to use named registers. Specifically, for those reasons, we need to be implementation-compatible with GCC when it comes to inline asm. So, whether we like it or not, we have to have that part working.
Maybe I'm missing something, but in the previous comments, you mention __builtin_<register_name>, and in the examples given by others, the register name is given as a string. Also, there is some example where the register name is associated with a variable via "asm". All these options are not exactly equivalent, but they all come with some issues.
If a register name is given via a string, as in "register long a asm("rsp")", who will check the type of "a"? On PowerPC, "fpr0" is a floating point register. Would it be legal to have "uint64_t a asm("fpr0")"? How about "float a asm("fpr0")"? What's funny here is that fpr0 is 64-bit long and is incapable of holding a 32-bit IEEE value. If you load a 32-bit fp value into it, it will be automatically extended to 64 bits. With VSX things are different, and if I remember correctly, it is now possible to have a single-precision values in some set of registers. Do you want the front-end to deal with all this? Actually, what's even funnier is that the official PPC assembler syntax does not define "fpr0". It's just 0, and the meaning of it depends on where it's placed. But I digress...
Another example: on Hexagon you can use pairs of registers, for example r15:14. Some instructions can use 64-bit values given in even-odd pairs like that. At the same time, you can use r14 and r15 separately, and both of them will be aliased with the r15:14...
But mostly, the __builtin_stack_frame is, in essence, a special case
of the generic pattern of named registers, and gives us the same level
of guarantees, so, in the end, there isn't much *technical* difference
in doing one and the other, with the added benefit that named
registers are already widely used and you won't need to add ifdefs for
new compilers.
__builtin_frame_pointer specifies a register via the functionality, not by name. In that sense, it is actually something more general than named registers. While many architectures have something like "frame pointer", not a lot of them have "rax".
Argument against it..? "Why?"
Why? Because the less asm and more C code we have, the better.
Yes, but making it "more C" by keeping direct uses of registers and only changing how that is accomplished, is, ahem, akin to porting code from C to C++ by changing the file name from .c to .cpp.
Because
inline asm semantics is a lot more obtuse than named registers.
I'm not sure if that's true. The compiler still needs to conform to the user's desire to have some some value in R3, even if everything except R3 would be a better choice.
Because builtins are just special cases of named registers. Because
target-independence is not possible in that kind of code.
Ok, so by now I'm confused. What exactly are we discussing here:
1. unsigned a asm("rax") : make a be an alias for "rax",
2. a = __builtin_register("rax") : copy "rax" to a,
3. a = __builtin_rax() : copy "rax" to a?
Is it about which of the above is superior to others?
As you can see, I don't like any of those, but some of them I like even less than others. 
The option 3 makes it easier to type-check, since each target can define its own set of builtins with proper types. On the other hand, what do you expect to get? The value of "rax" at that particular place in the code? How would you control what is being loaded into rax?
-Krzysztof