@Zizka Hm… it’s true that Stack Exchange is a little more of a Q&A forum where people attempt to compile knowledge without repeating threads on the same topic. Still though, no, I’ve never thought about it. I usually go there to look for answers as a guest reader. Also, a lot of my knowledge comes from dictionary trawling right after I receive a question, so I don’t know as much as I seem to know. Most of the stuff I say is a combination of my understanding of Japanese and what I read in a Japanese reference source that I stumbled through five minutes earlier. In fact, here’s an example right now: I’m gonna try to answer your question on だって with the etymology. (I sometimes wonder why I get myself into this when I could just give the most common uses and translations.)
スーパー大辞林 says this: だって is a combination of だ and とて, which itself is a combination of と and て. So, first of all, what is とて? It can be a case particle (indicates grammatical function), a linking particle (something that changes the nuance of a sentence ‘topic’, like は or も) or a conjunction. However, that’s all very technical, and I myself don’t understand all the implications of those terms… but in essence, all the meanings I could find for とて are essentially quite similar to と言って(も) or でも. What happens when we add だ? Well, mostly, it’s still basically でも, except for three other uses:
- Listing similar nouns, in which case it’s a bit more like も e.g. 私だって、あなただって、みんな日本語を勉強しています。
- Quoting what someone else said, usually because it’s surprising e.g. 何だって? (‘What did you say?’) 「六」だって? (You said ‘six’?)
- Justification (usually when it’s at the beginning of a sentence) e.g. だって今日はデートだから。(‘Because [I] have a date today.’)
In your sentence, it’s more like ‘even’ or でも. ‘Even about what happened during the time I was on duty,…’.
As for こだわって, yes, you can see it as sequential, since he was stuck on/fussing over his ‘failure’ first, and was only reminded of the patient when the postcard came.
EDIT: I think it might be easier for you to understand the link between と(いって) and all the functions listed above by remembering that 言う can also have a hypothetical sense in other languages, and its function is something like bringing something up for consideration (e.g. ‘Let’s say’ in English and “Disons” in French). In short, the Stack Exchange answer is more historically correct. However, your interpretation isn’t too far off either, since one of the equivalent structures for だって in my dictionary is 「…であっても…だ」, and if you remember that だ comes from a verbal simplification of である, you’ll see that it’s not too different from だって.