That’s well put, though of course it doesn’t apply to all imperfects, in fact only to very few; and I wouldn’t use “punctual” of any imperfect. The tense distinction in terms of “process” (impf) and “event” (aor.) works well, it seems to me, though linguistically informed discussions of aspect have given it up.
I don’t know how many mothers would agree that giving birth is a punctual action. 
It applies only to the imperfects that specifically the subject of this thread. The proper linguistic term seems to be “imperfective pro perfective”, although I don’t know how firmly this term is established. But I think “process” doesn’t describe the impf προΐει in the next example at all, προΐει describes an event, just like the aorist βάλεν. I think this is the way tenses are usually used when missiles are shot in Homer - they are launched with an impf and they hit with an aorist.
Il 3.346 ff.:
πρόσθε δ᾽ Ἀλέξανδρος προΐει δολιχόσκιον ἔγχος,
καὶ βάλεν Ἀτρεΐδαο κατ᾽ ἀσπίδα πάντοσε ἴσην,
οὐδ᾽ ἔρρηξεν χαλκός, ἀνεγνάμφθη δέ οἱ αἰχμὴ
ἀσπίδ᾽ ἐνὶ κρατερῇ: ὃ δὲ δεύτερον ὄρνυτο χαλκῷ…
But I guess “punctual” isn’t a very good term, I think “bounded” or “delimited” is better. The point isn’t how long the event took, but whether the narrator sees is as a single whole or not - I think this definition will suit better even for all those mothers… 
The “imperfective pro perfective” imperfects are strange, because we clearly see it’s a case where the action is a clearly delimited whole, and yet an imperfect is used, not an aorist.
“First Paris proceeded to launch his spear, Paris was the first to set about launching his spear.” It’s not an instantaneous act, it’s the entire process of pulling the spear back and and moving his arm rapidly forward and releasing the missile. Think a pitcher in baseball (or a bowler in cricket, if that’s more your thing). _Un_like the hitting its target - bam.
Similarly with ornuto of Menelaus’ turn in the last line you quote. So the thread gets back to where it started.
“Seen as a single whole” yes, but it has extension, unlike the aorist. I’d resist “pro perfective,” which I don’t think is right. Sure, the act is understood as having been completed, but that’s only implicit, context-determined. You could imagine a case where the act was thwarted (e.g. proiei, but he stumbled in the process and the missile fell harmless to the ground).
Or that’s how I see it.
Perhaps you’re right about that one; I think the question is more how to translate this than what the Greek is really saying. Whether call it “pro perfective” or not, it’s just us. Anyway, I think the essential reason to use imperfect in that particular context is to connect the action to what follows.
But can you explain how the action described by τίκτεν could be interpreted as something else than a single delimited event in the past? “Glaukos proceeded to conceive Bellerophontes”?
Iliad 6.152 ff.
ἔστι πόλις Ἐφύρη μυχῷ Ἄργεος ἱπποβότοιο,
ἔνθα δὲ Σίσυφος ἔσκεν, ὃ κέρδιστος γένετ᾽ ἀνδρῶν,
Σίσυφος Αἰολίδης: ὃ δ᾽ ἄρα Γλαῦκον τέκεθ᾽ υἱόν,
αὐτὰρ Γλαῦκος τίκτεν ἀμύμονα Βελλεροφόντην:
In many contexts ill is a synonym of sick that’s true. However, of the diseases that commonly afflict people in Britain most of the symptoms are difficult to delimit timewise (general aches, fever etc) but to vomit is by its nature punctual and for that reason seems odd.
“I was ill last week” is in no way an innovation, that again is true. It is the continuous tense that is the innovation. If English was logical then people would use the continuous tense when they want to say “Last week I was ill” just as they do say “Last week I was working”. And yes this does mean that “Last week I was sick” is ambiguous. It could mean “I was ill” or it could mean “I vomited”. If we look at a language expecting to see it to conform to neat linguistic caterogries then this is very messy. Looking at language historically then for the present form of a language to be shaped by the past is exactly what we should expect.
Now to bring things back on topic.
Sure, punctual is not a very good description of the aspect description expressed by the aorist-imperfect distinction. We are used to the idea that whether something has delimited aspect or not is a matter of choice. An event that lasted several million years such as the end-Permian mass extinction is, from the geological perspective over in a flash. A bullet slamming into a block of wood can be by us watched as a slow open-ended process by simply playing a film in slow motion.
However, in a language where two verb forms are changing in the extent of their usage then a perfective/aorist form will be retained for punctual events even if it is not used in other cases where we might expect a perfective/aorist form to be used.
I say “retained” but I suspect that if Homer tends to use the imperfect in cases where later usage would lead us to expect an aorist then that is because the aorist was in his day the new kid on the block and was in a process of expansion.
Excellent! It’s a perfect parallel with that λιπε - λειπε sequence in Il.2 of Agamemnon’s sceptre, isn’t it? A did X, then B proceeded in turn to do Y. It’s significant, as others already suggested, that X (aor.) is the background event, plain and simple, while Y (impf.) is stayed with (both the scepter and Bellerophon), so that (to adopt your formulation) the effect outlasts the action. (The reverse, in a way, of your προιει … βαλεν example, where an action is set in train with the impf and its result is a simple event.) This use of ετικτε, as I noted above, is common in tragedy; it seems to mean more or less “was his/her mother” – or father, in this case.
Actually tense usage remains pretty consistent throughout the history of ancient greek, from Homer through late antiquity. The aorist was no new kid on the block (with or without augment – there’s controversy over whether this was older or newer). In Homer the aorist is in constant use and highly developed (you have all those variant forms, elabe elaben ellabe ellaben labe laben lab’ - and gento to supply the metrically missing one), and when the imperfect is used instead it’s for a purpose, just as is the case with later writers. If we sometimes have difficulty seeing the point of an imperfect (or any other tense), that’s just a reflexion of our inadequate attunement to greek usage. As Markos commented earlier, “the Greek FELT these distinctions.” I make so bold as to say that we can too, if we train ourselves to develop sensitivity by reading with close attention to tenses and constantly asking ourselves why this particular tense is used, and what difference it would make if a different tense were used (aor. vs. impf, for example). (The difference is always semantic, I’d say, and so do readers far more expert than me.) That way we become better readers.