Split up C++ diagnostic emitter and factor it out. (#7558)

Split the diagnostic emitter into a separate emitter (regietered with
Clang) and listener (registered with the emitter). The purpose of this
split is to make the Clang emitter not depend on the `Check::Context`,
so that we can use it, and hence the same Clang instance, with multiple
`Check::Context`s. A fallback listener is registered to collect and emit
any diagnostics produced while we don't have a `Check::Context`
registered with the emitter.

Assisted-by: Gemini via Antigravity

toolchain/check/cpp/context.cpp

@@ -18,6 +18,11 @@ CppContext::CppContext(clang::CompilerInstance& instance,
 
 CppContext::~CppContext() = default;
 
+auto CppContext::set_diagnostic_listener(
+    std::unique_ptr<CppDiagnosticListener> listener) -> void {
+  diagnostic_listener_ = std::move(listener);
+}
+
 auto CppContext::clang_mangle_context() -> clang::MangleContext& {
   if (!clang_mangle_context_) {
     clang_mangle_context_.reset(ast_context().createMangleContext());