1. As you'd pointed out, the language in question might not let you generate certain instructions. For instance how do you express LGDT in C? You can't. You'd instead write it in assembly (or inline assembly) and expose it to the C world as a function (or in a compiler that supports it, an intrinsic) that it can call.
2. A CPU's understanding of state is defined around the notion of registers and memory. A programming language however exposes a higher level abstraction. For instance, a function call in C is an abstraction that defines both transfer of control (which the CPU knows about) and persistence of local state (saving and restoring registers, stack pointer etc., which the CPU knows nothing about). This is called the ABI and is merely a previously agreed upon convention. Higher level languages (say Java) are able to provide richer abstractions than merely an ABI as their "agreed upon conventions" are defined at a much higher level than in terms of a CPU (registers and memory). Java, for instance even defines a virtual architecture in terms of which facilities offered in the language are defined. User code written in one of these higher level languages depends on these facilities being available to them at any time. This is what makes it unsuitable for writing certain sections of OS code. For instance, the CPU expects the OS to save and restore register state around an interrupt handler. If said handler was written in C (as a function) though, it'd expect its first six arguments in certain registers etc. - conventions, the CPU knows nothing about. You can think of the CPU as exposing its own, simple ABI which can only be implemented by writing certain sections of code in assembly (This is not entirely true though, as one might be able to mimic something like this in certain compilers, especially C compilers, that let the programmer write "naked" function bodies and custom prolog / epilog code). This is also why most parts of the OS, that don't directly interact with the CPU, can and are written in a higher level language. For e.g. The Singularity OS by MSR was written in C# and assembly.