There is a major drawback to save files which rely on the deterministic nature of the game: what happens when the determinism changes? I don't mean due to bugs; I mean intentional modifications to the game's behaviour through patches. Either these changes will break old replays and saves, or the game needs to carry around multiple simulation versions.
You bring up Age of Empires. This is a small problem in the current AoE2 competitive scene. Yes, AoE2 has a competitive scene – the game has been re-released twice in the past decade, and still enjoys the attention of competitive play, including sponsored tournaments (e.g., the Red Bull Wololo series). Microsoft are attempting to give the game the same treatment as other modern, competitive titles, which means frequent balance patches, and occasional injections of new content (DLC). Unfortunately, each one of these updates breaks old save games and replays, which, as you point out, rely on the deterministic nature of the engine. It means that any casting/commentating of games must happen within relatively short order of them being played, and that the only reliable distribution format for old games is... screen recordings. I'd much rather a less space-efficient save format (or at least having the option to convert deterministic replays into a fully self-sufficient archive format) than having to discard native recordings of great games by the wayside in favour of videos.