The example given by the Man in the Funny Hat sounds to me like trouble has already started, but both sides give the other a chance to back off. Mechanically speaking, initiative has been rolled, but both parties have decided to delay on their first turn (or do anything but attack, anyway). That way, when one side decides to take an action other than delaying, nobody's flat-footed.
I have actually used initiative rolls for things other than combat in the past - and sometimes these situations devolved into actual combat after a moment. In these instances, flat-footedness didn't come up, because everybody had already acted.
It's been my experience that using initiative for anything other than combat, or even using it before someone ACTUALLY attacks only leads to the realization that the initiative rules break down when they "exceed their authority." Again, if two characters are arguing and you expect it to break out into an actual combat at any moment and you want to be sure you know who's going to go first so you have everyone roll intiative. But then combat
doesn't happen and there you are conducting combat rounds when there is no combat. Then you find that combat doesn't actually start until it's somebody's turn - but how do you determine then who gets to start? That is SUPPOSED to be what initiative is for.
I don't think the rules actually say it but combat doesn't ever start until there is an attack. Spell, sword, special ability - just something that assaults another character or creature with an intent to harm or control. It also doesn't start with an effort to defend yourself. You can take actions to bolster your defense but that doesn't start combat and thus does not require initiative. Initiative is rolled when somebody says, "I
attack". Initiative is only used and only needed when combat is actually underway. Prior to that you can use whatever means to sort out actions that you think best. Now initiative is a Dex check - but making dex checks is not always rolling for initiative.
Now here's something else that I don't think is actually in the rules, but the ONLY thing that initiative does is randomly determine the order of action resolution when you don't already know who goes first. Now there's nothing quite as silly as having a player state, "I attack," and then after initiative is rolled end up going LAST. IMO, in any encounter when surprise is not a factor and the DM determines that there isn't an immediate, general consensus to fight, then the first person to say, "I attack,"
wins initiative. There is no Ready action outside of combat. If you want to draw your weapon you say so and your weapon is drawn. If you want your character to be the first to fire or attack when combat has not yet begun then all you have to do is be the first to SAY, "I attack." Then everyone else rolls initiative.
As I said, initiative is simply for determining the order of resolution for actions. In its finer workings it doesn't even determine who actually goes FIRST - just the ORDER in which actions are resolved. When someone says, "I attack" the question of who is at the top of the order becomes known and shouldn't (can't!) then be re-determined randomly.
All of this simply argues that the rules for initiative are well-written even if they don't go into detail about why they apply to various situations the way they do. Use initiative as written and only when it is specifically called for and it works. I'm a big believer in DM adjudication, and ignoring or altering rules that get in the way, but in the case of Intiative keeping things as BtB as possible works best.
JMO
