[app-server] Make sure that config writes preserve comments & order or configs (#7789)

Make sure that config writes preserve comments and order of configs by
utilizing the ConfigEditsBuilder in core.

Tested by running a real example and made sure that nothing in the
config file changes other than the configs to edit.

codex-rs/Cargo.lock

@@ -851,6 +851,7 @@ dependencies = [
  "tempfile",
  "tokio",
  "toml",
+ "toml_edit",
  "tracing",
  "tracing-subscriber",
  "uuid",