Two big advantages are referential transparency and memory aliasing. The sort of reordering of loops/memory/whatever like described on the Halide page is just the beginning of the kind of optimizations that are possible. Perhaps static analysis could ameliorate this to some extent, but it will take far more computational power, and will always be subject to the halting problem. There's also current problems with the C standard that could change in a new standard, or maybe optionally ignored in the compiler, so it depends on your definition of "C". Examples of this are struct member ordering being fixed, or lack of tail call optimization (at least guaranteed TCO), and perhaps you might include calling conventions in that.
There's also the huge advantages you can get in programmer productivity. Metaprogramming, better type systems, better (and more) abstractions, automatic memory management, automatic parallelization, there's really tons of room for improvement in so many areas. And a lot of these provide lots of room for optimization, in the sense that they put more of the program semantics into the compiler's hands. Compilers will probably be "stupid" for a long time, but at the very least they are very accurate and very fast, so they can try out many possibilities and measure what works best. Chess is a good analog here--computers are far superior to humans these days, and they got there by just trying tons of permutations of simple algorithms and running lots of tests. Writing compilers turns out to be a similar problem, so moving away from C/C++ can have advantages at multiple levels (so meta!).
It's still in a very nascent stage, but the language I've been working on should eventually be the manifestation of lots of these ideas: https://github.com/zwegner/mutagen Most of the ideas are just in my head at this point, and still rather fuzzy, but there's some specifics in the README about possible future directions.