Sadly I do not have the access to Japanese Yahoo. But I understand what you mean, that is why I disagree. A sentence can be left hanging only when something comes after it. As you stated, it is hard to find anything on the Internet regarding the full phrase that どういたしまして comes from, or if it comes from anything at all for that matter.
The difference that I’m trying to point out is that the implied meaning is a paraphrase of the phrase どういたしまして, not it’s full version, nor something that is being omitted. I think we can compare it to idioms, for example “beat around the bush” means “not getting to the point”, but the meaning, while implied, is not omitted in the original phrase - it just isn’t there, physically for lack of a better term.
I think the same goes for どういたしまして. It has its meaning, everyone knows it, but it’s not physically there.
The way you use the て form as you are proposing (and as far as I know) is like this (my previous example)
すまない、忙しくて…（来られなかった）
You take a compound sentence, leave out the part that is obvious, and end it with て.
Now why is this so important to me you’re probably asking, it seems extremely trivial right? Why don’t I just agree with you that yeah something is being left out there and you can use universally this one interpretation of て in every single case. It’s because I’ve seen examples where this interpretation makes no sense and I’ve been left unsatisfied with my understanding of these sentences.
If you care, you could explain to me for example this (from the wiki):
ちょっと売れたからってすぐ天狗になって。なんだ人を馬鹿にして。
Or this:
私、市役所の担当の者でございまして。いまご近所を回ってご説明させていただいておりまして。
Why do these sentences end with て and not regular non-past or past. What further information follows these sentences? It genuinely makes no sense to me.
@CursedKitsune I agree a native speaker does not think about the grammar they use 
But as you can see I am not native, and this gives me trouble.
Edit. Small clarification of what I want. When you look at すまない、忙しくて it’s pretty easy to imagine that something is ommited, at least to me. I don’t get the same feeling with the above examples. If you tell me to take it at face value, then even if I agree, that doesn’t change the fact there is a missing piece in my understanding of this. Now of course you have no obligation to teach me but if you can, I think more people would benefit.