Switch custom error stream output to diagnostic (#4846)

This switches most error printing to use diagnostics instead of direct
stream writes, even when not a specific file diagnostic. I'm allowing
empty filenames for this use-case.

This allows a little more specific testing to validate coverage of
output using the diagnostic coverage test. I'm adding a few tests to
cover things that weren't previously tested.

Separately, this also forces a little more standardization in format...
considering how changes like #4568 show effort being spent to _mirror_
diagnostic style, my thought is now to just use diagnostic code where
possible.

Note this also allows incrementally better testing of the language
server; I'm changing the crash fix from #4847 in favor of diagnostic
testing.

---------

Co-authored-by: Chandler Carruth <chandlerc@gmail.com>

toolchain/check/check_unit.cpp

@@ -366,7 +366,7 @@ auto CheckUnit::ProcessNodeIds() -> bool {
     auto converted = unit_and_imports_->unit->node_converter->ConvertLoc(
         node_id, [](DiagnosticLoc, const DiagnosticBase<>&) {});
     converted.loc.FormatLocation(output);
-    output << ": checking " << context_.parse_tree().node_kind(node_id) << "\n";
+    output << "checking " << context_.parse_tree().node_kind(node_id) << "\n";
     // Crash output has a tab indent; try to indent slightly past that.
     converted.loc.FormatSnippet(output, /*indent=*/10);
   });