Why is there a need for an always present button to configure the mod as well as a hotkey? Why not use the regular mod settings menu for it instead?

You most likely never need to change it after installing the mod, never mind on the fly. I could see it being useful at most when configuring it for the first time, but after that? Having it always take up space is kinda pointless.

you've likely got the wrong mod, this mod has no ui button, most likely the correct one you are using is https://mods.factorio.com/mod/Hazard-Lights-Redux
my mod here is currently abandoned, and also it seems there's some way to remove the button from that newer mod as well

Oh damn, my bad, I didn't realize XD. I kept yours too in the hope that you will eventually update it, thanks for clarifying.