Forge has a feature for displaying notifications on mods what are out of date. Could this mod also use that, at least in the main menu? I am unsure of whether it would work whilst in-game.

Display like this?

Ye.

https://www.curseforge.com/minecraft/mc-mods/gamemenumodoption/files/3273863
Added update notification function.

I do not know why you think that, nor what (Mixin) coremod means.

I have a strange feeling that the thing you've implemented is not exactly what they asked for. And because of that, your nice mod has now become a (Mixin) coremod :/

I don't know what you meant, but this sentence: "Could this mod also use that, at least in the main menu?" makes me feel like you wanted the green gem back on the "Mods" button in the main menu, but:

1. It's a Forge thing implemented by its developers a looong time ago, then at some point they dropped it (probably during the huge 1.13 rewrite), but now it's back again for at least several months. No need for any third-party mods here.
2. Not every mod supports this feature (I think most don't) as it is an optional entry in the "mods.toml" file and requires some extra work from the mod developer. Also no need for any third-party mods here.