I didn’t watch the movie, so I’m probably missing something. Did Georgiou also have to deal with a bunch of normal assassination plots after officially gaining the throne? Or are we to assume that by virtue of winning this contest, she is widely seen as too dangerous for anyone to attempt to usurp?

If the TOS Mirror universe is anything to go by, yes. People are waiting around with knives and daggers around every corner. If you’re in a position of power, you have to be prepared for assassination by anyone and everyone. Lorca didn’t sleep with a phaser under his pillow for fun. That is a problem that almost certainly gets worse the higher up you get in the empire.

Being Emperor just means that she has a lot more tools at her disposal to fend off such things, or make enough of a threat that most would think twice. Mirror Kirk had his dimensional machine that basically evaporated people by sending them to another dimension.