Static analysis helps, but it can't catch everything. I work on a modern C++ codebase, and we still face all of these issues.
Formal verification is infeasible for most software projects, but they can get guaranteed type/memory safety by using a language proven to be safe. C/C++ can't give you that, but JavaScript might be able to.
Things that are written in C these days are usually written in C for performance reasons. FFMPEG would not have even close to the performance it has if it was written in a memory safe language instead of C and assembly. I doubt that a magical compiler (and/or language) will appear in my lifetime that can compile high level code into performant machine code, especially when it comes to memory management. (note that C also has advantages other then performance)
JS doesn't even have a proper specification, let alone a bug-free interpreter/compiler.
EDIT: AFAIK verifying memory access is part of a formal verification, where memory is also modeled mathematically.
So is C the problem, or is it modern CPU architecture? C has stuck around for so long because of how close it is to assembly language. There will always be a need for a language that is one layer above assembly, and currently assembly is incredibly hard to secure.
C is close to PDP-11 and 8/16 bit computer Assembly, it has hardly any direct mapping to modern CPUs.
Back then the compilers sucked. They would take complete crap of code and still it would work. They were like browsers are today. (from my experience from going through one old MUD code)
Today the song is different. Not only will the compilers warn you of many things, there's even tools for static analysis (and dynamic). So the argument that C (and even the more complex C++) is inherently insecure holds much less weight (just go run old code through a static analyzer, or a normal compiler for that matter).
That said there's only one way to write a "secure program", and that is formal verification.
People that talk with a serious tone should back up their claims, at least that's my opinion.