Thanks! I still don’t really get it, but telling me I need the Lagrangian formalism is helpful, and tells me what I need to study in order to really understand Noether's theorems.

I’ve heard of the Lagrangian before, but my knowledge of physics is probably somewhere around the level of someone who has only just finished the first year of their degree, at least judging by the modules my alma mater lists for their physics degree (I got Software Engineering from them 14 years ago, all my physics knowledge since then is personal geekery).