TE form by itself has never been a "polite request form." But it's how you form polite requests such as "~TE KUDASAI."
That's true, just the way I originally learned it made me think using the て form in itself (when used alone) caused the feeling of a request, and I eventually found that to not really be the case...
It's not just implied though. These phrases are part of the language and well understood.
The word I should've used is "omitted" as
@Toritoribe worded it, as in, the sense of it being a request comes from the omitted word that comes after て, and the omitted word is "imagined" due to the fact that as a connective form, using て in the first place implies the speaker intended to connect something to it, but omitted whatever would've been connected. Basically, my feeling is that it's only understood because of the fact that it's expected for something to be connected to the て, as the connector it is. Hope that makes sense.
Indeed the -te form can be an imperative, but it's not a polite imperative but more likely a casual soft imperative. Thus, you can't use this form to your superior.
I guess I just never made that distinction before, I thought only the imperative form (ろ、れ、等々) could be considered imperative, so I considered て as a "request" the whole time... And I guess in contrast to how the imperatives sound, I considered that polite （and also with how tightly wound the て form is to ください and other 丁寧語）
This is not the contraction such like なければ vs. なきゃ, but the omission of the following word (e.g. 食べて vs. 食べてください) since it's obvious from the context.
I used a bad example I guess, since as you said, it's a contraction, but omission was what I meant.
That's exactly the fundamental function of the -te form, therefore, it can show, for example, the reason/cause.
Now that you said that it helped me make sense of times when people end their sentences with て
↓
teacher: なぜ遅刻したの？
student: 事故で電車が遅れて。
With this example, could I be right to assume to technically "finish" the sentence, the student would've most likely said 「遅刻しました」? Now it makes sense, it's just like ので in this case
It all clicked now, thank you guys
So basically, て is the connector, and whatever could be expected to come after it gives it meaning, whether to request or to give reason.
One more thing I have to ask, which is about the formatting of the original question:
When I typed 「何何て」, I knew I felt something was wrong with it and kept switching between 「○○て」 and 「何何て」, but after reading my question again, I finally figured out why I felt that way.
At least for me, having て right after the word なに sounds very unnatural, so I think if it were a Japanese person posing this question, they might've used 「何何って」 instead. There's probably a more natural way altogether to make the question, but am I right in feeling this way? If て is right next to 何 it feels like it has to be read as 「なんて」, but the two 「何」 indicate 「なになに」 so it feels contradictory and just outright wrong. I thought I would write it without っ as that's usually a property of the verb itself and technically not part of the て form. Would ○○て be better (despite the censorship connotation it has sometimes)?