a bunch.
you can't play whatever you don't know and you cannot identify what you have not seen.
@1
"how much effort do you really need to put in to familiarizing with a variety of openings to make progress?" ++ None at all. Opening study stands in the way of progress. Real progress is to gain the ability to find good moves also in the opening without depending on memorisation. That same skill also helps in the middle game when you are on your own anyway.
@2
"Modern grandmasters know everything inside out"
++ No, they do not. That is why they sometimes dodge it, like playing the French Exchange Variation: twice in the Tata Steel Masters.
I know a lot of advice is to focus mainly on tactics, positional strategy, and to really drill one or two opening repertoires. I've also mostly done this lately; however, I've also found that especially in any short time control you are at a serious disadvantage when someone plays some novel opening outside of your expertise and they pretty much spam it so they know a ton of traps in it, the main line and several variations, the motifs, themes, and ideas. Yet, there I am trying to figure out a proper response and avoid getting hit by traps while my opponent just blitzes out their moves without hardly even thinking.
There's also the cases of some of the most popular openings like the Sicilian that you don't really get a choice whether you're playing against or not, and have a gazillion sharp variations. So I'm just wondering from the expert players above 2000 -- how much effort do you really need to put in to familiarizing with a variety of openings to make progress?