TL;DR: The behavior you describe is what Gene Rodenberry would have liked to believe Starfleet would behave like; but we can surmise that's not what actually happens in that universe.
It's Federation propaganda. To see why that is, let's take this issue out-of-universe first.
Remember Pericles' famous Funeral Oration, where he describes the virtue and magnanimity of the resplendent Athens? Or the frequent speeches by US presidents explaining how they strive to bring peace and freedom to different parts of the world? Well, that's propaganda: Pericles was busy building his long Pirean double-wall, planning for war after he makes off with the funds of the Delian league. And the US invades, changes regimes and bombs and kills millions to further its domination and its ruling class' economic interests; it is hated and feared for this reason around the world. But no empire describes itself, imagines itself as ruthlessly applying naked force - they tell their story differently.
Gene Rodenberry aimed to describe an idealized humanity in space; but the dominant ideology in the US shows up in his creation - as no one is unaffected by the processes of socialization (or inclucation) of his/her surrounding society and the state governing it. And while Rodenberry did not intend Star Trek to be propaganda, the forward-projection he made (and was expanded and extended by the rest of the creative team) carried both good and bad (if I may use such terms), both what is consciously acknowledged and what is glossed over, swept under the rug or flat-out denied.
We thus find the Federation to be a militarist Human-species-supremacist command society, which is mildly bellicose and more than mildly expansionistic. I won't go into the details of that analysis here (you can follow the link), but suffice it to say that the broader facts we learn from the series (especially TNG and DS9) support it quite well, albeit through the resolution of contradicting evidence. With that being the case, you can rest assured that Star Fleet, like any navy/military, does not typically wait until the absolute last moment before firing; and it is not led by poet-philosophers captaining the ships (Plato, anyone? Yes, another authoritarian strand) like Picard or Janeway. If they don't actually shoot first it's probably because they're hacking into the other ship's system or have a wider political machination in play which calls for avoiding combat.
The thing is, we (whether in the US or not) would not cheer for this kind of protagonists, for that kind of behavior. It's not because there would be no story - you have quite a few interesting episodes about the Klingons or the Romulans, or the Cardassians, who do - supposedly - shoot first. It's just that viewers would not tolerate (justifiably so, I suppose) that from those characters with whom they're supposed to identify. So a different yarn is spun.