Hey Isaac! Good question. I don't think there's an answer per se. I'd say it matters regarding what you're using it for and what your goals are.

It's just a matter of what you work with and what you want to hear better. I think in terms of the specific genre sounds it may make more sense to be a bit specific - like if you want to work on vintage trance maybe it doesn't make as much sense to work with the indie folk packs or whatever (not sure if there's an indie folk pack btw off top of my head).

You'll typically use EQ on everything.. so EQ makes sense to work on everything you want to work with... so if you use a kick drum, yes to kick drum, if you use acoustic guitars, yes to acoustic guitars, if you do vintage trance, yes to vintage trance haha etc.

With compression you also typically use it on everything to varying degrees. Not saying everything needs compression, but there isn't anything specifically exempt from needing compression once in a while in certain circumstances. (Though I think they only have drums and perc because from my understanding compression works SO differently on different sources and with different settings that it would be really hard to make a game from it for all sources unless they gave you the exact settings used perhaps. This is why they say at the start that it's fast attack and release times, because slow attack would kinda have the opposite effect lol. It'd probably be bordering on impossible to hear the super subtle compression that they have on higher levels on nontransient materials such as the smoother strings for instance, or vocals. I'm on level 29 in Dr Compressor and it takes me 3-5 minutes sometimes to just hear the subtlest difference in the drum compression they're using. I doubt I'd hear that AT ALL on something with little to no transients lol)

Reverb is also used on pretty much everything at least once in a while... not sound specific. It's often used in smaller doses than say EQ, but the game helps especially when it comes to subtle reverbs and hearing that a signal may still sound dry, but it's not actually dry when you compare it - so it may not sound like others are using reverb, but they actually use a bit to make things sound a bit more natural without sounding reverby per se.

Normal to pan everything too... to varying degrees.

Delays you usually use a bit more sparingly than EQ and Compression, and likely a bit less than reverb too. (But of course depends on creative decisions etc.)

Distortion would definitely depend on the genre, and it's good to hear in general to avoid distortion when you don't want it, and to also know what saturation adds to an instrument or mix. It's been helping me hear things like subtle tape saturation frequencies etc a bit better. Again, works for everything.

So literally all the effects are used on all the things. The things you use depend on what you produce or mix etc... so I'd pick based on that.

Sure there are some advantages to hearing effects better with certain sounds, and will help advance levels quicker, but it may not always serve you in the real world as much.

That said though, the higher levels in my opinion get SO specific and subtle that for practical purposes it won't make much of a difference. In my opinion it's just to push your hearing to be even better and more precise, but in reality a difference of like a hair of extra distortion where it just lightly compresses the signal in a barely noticeable way isn't going to make or break a mix or production. I think it's just to refine the hearing so that you can work better and more efficiently with the effect and know what to listen for.

One additional thought: with EQ mirror and EQ knight, it definitely helps to have something that has a broader frequency band such as full songs. Sometimes for instance the settings are above 8khz or below 200hz and the particular sound doesn't have any information there, so it gets annoying. However this experience also helps to learn that this may be the case and how those boosts or cuts will actually affect such a signal (sometimes you hear a very subtle change in neighbouring frequencies for instance). But with those games I'd pick full mixes/songs usually, or sounds that have wider frequency ranges. Asides from the songs I think the strings are pretty broad, the piano can be, the drums too. Trumpets are hit or miss. Vocals sometimes lack super high freq and the super lows.

Anyways, hope this helps.

TLDR version: whatever works for you, no actual answer. All the effects are used on most things, it just depends on what you work with and what you want to work on.

Also, at the end of the day, it all helps your ears, so the skills learned translate regardless.

I think the best answer I can give is trust yourself on it - because you know what's useful and challenging, and what's easy but not useful. You said pink noise sounds a bit like cheating, and though it isn't cheating, I agree with you. I never use pink noise in EQ mirror because never do I really have to EQ pink noise... and that doesn't help me. Maybe I'll try it to get some points lol, but it's not the best motivation.

A kick drum in distortion would probably be a bit easier than the acoustic guitar, especially because it has such a clear transient and it's a bass sound, so you can hear higher frequencies being added a bit easier. But is it that beneficial to advance many levels on just the kick drum? Or to switch it up?

However if you're a Kick Drum only audio engineer (; (probably doesn't exist - super specialized!!) then yeah I'd keep working on it.

Or if you're just a drummer, or a drum mixer, then it may make more sense to spend time on the kick drum and the drums rather than acoustic guitar.

Anyways, I don't think there's an objective answer. It's subjective. So trust your own subjective learning/challenge/usefulness.