Patterns in auxiliary verbs (機能上の補助動詞)

I’ve been having trouble with auxiliary verbs (補助動詞). Even though they’re extremely common, I’ve been having a heck of a time guessing them from context. What’s more, they seem to have a large effect on the overall thrust of a sentence.

Japanese has two main groups of auxiliary verbs. I translate them as formal auxiliary verbs( 形式上の補助動詞) (the ones that come after て) and functional auxiliary verbs (機能上の補助動詞) (the ones that come after a verb stem).

I personally have more trouble with functional auxiliary verbs.

The one that bugs me most is つける・つく (more precisely, the several verbs that have this reading). I think the つく most common in auxiliary contexts is 付ける・付く.

The verb 付く has a dozen definitions in Jisho but it seems to generally mean “to attach.” I try to imagine something being glued to something else.
追い付く: I try to think of someone chasing me and getting so close that they got a hand on me. And the hand is covered in glue. And now we’re stuck together.
思い付く: my brain is flypaper and ideas are getting stuck on it.
傷付く: I got a wound stuck to me, like one of those movable holes in Who Framed Roger Rabbit.
寝付く: Sleepiness is sticking to me. I wish it would for once.
噛み付く: Something bites me and its teeth won’t get out of my skin. This one’s easy for me. I have three cats. They hate tummy rubs and yet they have tummies. Curious.

Unfortunately, 着く also appears in plenty of compound verbs, so sometimes I have to think of “arriving” instead of “sticking.”
落ち着く: I arrive at CTFD. But then, 落ち付く is a variant, so I suppose I can also think of sticking calmness to me. Which my anti-anxiety meds are supposed to do. In theory.
辿り着く: I follow, I follow, I follow, and I finally arrive.

And then are also つく verbs where the kanji is 突く because this language hates me.
楯突く: In America, we would say “to bite the hand that feeds you” or “to shit where you eat.” In Japan, that’s apparently stabbing someone with your shield. Fortuitously, my girlfriend recently made me watch 盾の勇者の成り上がり, whose main character is a literal manifestation of this metaphorical verb.

Next up: かける・かかる. In functional auxiliary verbs, it seems to usually mean “to hang.” I try to think of a coat on a hanger, but most of these verbs don’t lend themselves to that. This one seems to arise in verbs that themselves have a lot of definitions and depend on the context, like 引っかかる and 差し掛かる.
御目に掛かる: “It was an honor to meet you, and by ‘meet’ I mean ‘dangle my eyes on you.’”
鍵が掛かる: This one’s weird. If a door is locked, and a key is hanging on the lock, that seems like a problem with an easy solution.

Next: よる. I know よせる is the transitive equivalent, but I see よる a lot more. It feels like random chance whether the tacit kanji is 寄る (“getting closer”) or 依る・因る・拠る・由る (“to be due to”) or another verb altogether.
近寄る: This one’s not too bad. Both verbs deal with closeness.
片寄る: I wish WaniKani had this one. I’d have an easier time remembering its more common variant, 偏る.
年寄る: To get older is to get closer to a year? I guess?

The fourth auxiliary verb I struggle with is 込む. The transitive version is 込める, but again, I see 込む more often. I picture some person/object bulging because something was inserted into it.
申し込む: To apply or make a request is to stuff words into someone else and hope the word “yes” comes back out?
突っ込む: You stab someone. They are now one knife fatter.
思い込む: Somehow a thought got stuffed into me?
引っ込む: I retreat and stuff myself into my… retreating place?

Thinking this out, I realize that part of the problem is that there are several “first” verbs that I still struggle with as well, including 引く and 差す and 割る. But that’s a topic for another post.

I know this is a long OP, but this has subconsciously bugged me for years. Besides, I wanted to make sure this isn’t one of those OPs where I lowkey ask you to Google for me. I was going to post this on Bunpro’s forums, but I thought this forum might be better suited. Maybe some of these have bothered you too and we can all help each other out. I just want to stop involuntarily checking out every time I try to read a sentence and see つく.