I have configured the webhook URL and set "Use Discord Webhook" to true but nothing gets posted to the Discord channel when using the MLRS, no errors get logged in the console either. Console logging works fine, though.

However, I did notice that on the plugin page it says "works with Discord Messages", so this makes me wonder the following question now: Does it simply work with it or does it actually depend on it?

For now I'm using Rustcord's custom logging function so the MLRS tracker console messages still get posted to the Discord this way.

it will require that plugin

It's an optional dependency, which is why it says "works with" rather than "required"; as it's only necessary if you want to use the Discord support.