I think a combination is ideal. You want to know the more common variations more deeply, and the less common variations not as deeply. I also think it helps to do multiple passes, kind of like a parser does.
So when you're first developing your repertoire you might intentionally limit yourself to depth ~5-7 depending on the lines commonality. Just to ensure you get the basic overview, and settle on a coherent repertoire you're happy with. And from moves 1-3 maybe you cover every line played at least 1% of the time (with varying depths of course). But for moves 4-6... maybe you don't cover moves that are played <= 3-5% of the time, with some exceptions. And for move 7... maybe you move this to <=5-10%. etc..
You can adjust these numbers however you see fit, but the point is you have some limit both on how deep and how broad you go, and it differs based on the lines probabilities of being played.
Now on your 2nd pass... which you don't do until you're very sure about the 1st pass - so you know exactly which repertoire you want to play and why... you just modify those numbers. So now you go to depth 10-12... And now you cover moves <=3-5% starting at move 4, and <=5-10% starting at move 7, and then <=10-15% starting at move 9. etc. and maybe you adjust these numbers depending on the line you're in as well.
Of course your repertoire will continue to evolve as you play it and change your mind or experiment, so this is not a rigid process... think of it more like growing a plant, but at least there is some structure to it. But you don't want it to go too deep too fast or you'll prevent yourself from being able to experiment or change your mind. But eventually you'll probably settle on lines you like. So I suppose you're prioritizing breadth in a way, at least at first.
It's hard to give perfect advice in the abstract though. Because, for example, as black I ignored everything except for my response to 1. e4 and 1. d4 for a long time, just due to the sheer size of the repertoire otherwise. Until eventually I added some lines vs. the english and reti, but I still only know the first 5-7 moves of those. Though bear in mind - the e4 / d4 lines you play influence how you deal with the english / reti.
On another note... I don't recommend the Najdorf at your level. It's so theoretical and a6 is a more passive move, meaning there's too much theory and when your opponent goes wrong during the opening... you don't really get to punish him hard the way you do in other sicilians, like the classical where... for example, white can't even play the english attack setup or you have an advantage immediately.
Should I be starting with depth or breadth? Is it better to know the 3 most common variations 20 moves deep, or the 10 most common variations 10 moves deep?
How much of my goal should be memorization and how much of it should be understanding? I could have a grand old time with the engine on for hours studying one variation, but that seems like a very inefficient way to learn an opening with so many viable branches.
Your suggestion?