I haven’t had time to read the entire discussion so forgive me for potentially making it even more confusing than it already is; I’ve also never really bothered to learn “individual uses” of the て-form so this is doomed to be an unhelpful comment already, but I can at least try to clarify the lines a bit.
Context: knowing it was useless and the downfall of society etc etc, 白鳥 (I missed the romanisation somewhere, is it Shiratori?) still let the patient get a useless surgery for money’s sake, at the professor’s orders. Saitou thinks Shiratori is a hypocrite and that he should have opposed the prof, but then comes to the realisation that he himself is essentially the same; he didn’t say anything back to Shiratori and “wasn’t thinking of the patient at all”… Then comes this phrase:
“I was only thinking about myself… Even on call I was only fussing about my own mistakes…”
だって corresponds to “even” in my translation. I’m not sure if it originally came from copula+て-form or anything like that, but in modern Japanese it’s used like でも (unless it’s copula+quotation particle って) and doesn’t have much of a relation to the て-form. Here are some examples of its uses, “1 〔…でさえ〕even” and “2 〔…もまた，…でも〕” are the ones that apply to this sentence.
Then about the て-form… The way I think about it, it has two uses: 1) to connect a verb and auxiliary verbs/grammar points (e.g. , 話している, 試してみる) and 2) to connect other stuff. “Other stuff” can be anything; words, sentences, events, different speakers, even nothing (trailing off a sentence). It doesn’t do much, except showing that A and B are in some way related to each other - how they’re related is up to you to figure out based on the rest of the sentence. Now that I’m typing this I wonder if you can translate this second use as “and”; might not always give a pretty sentence but maybe it’s easier to imagine the relationship between thing A and thing B that way.
In this case, what Saitou says next is:
“I’d completely forgotten about the patient until the letter came”
So the て-form こだわって means that Saitou was fussing about his mistakes and had forgotten about the patient until the letter. […]こだわっていた、そして[…] would mean the same thing but it’s longer and people are lazy.
Different examples for good measure: 頭が痛くて宿題に集中できなかった"my head hurt and I couldn’t focus on my homework" -> Both things happened so it’s not wrong to use ‘and’ per se, but most people will instinctively say ‘because’ instead.
電車に乗って会社に行った “I rode the train and went to the office” -> again, both things happened, and while I don’t explicitly say it that way, most people will understand I went to the office by train.