Others have given comparable examples of how this works but since you asked me I'll answer as well.
A parser giving you a 1245 will not tell you to when use BoTD by itself, in fact nobody would look at it that way anyway as that's a prime example of not correctly reading the information.
There's a large difference between doing the calculations when it comes to the potency of skills etc to create the perfect rotation before hand, practicing it ad nauseam on a striking dummy and then actually performing the rotation successfully in live content with the variables that are boss moves and other party members.
When players use the term "improving their rotation" they don't mean that the parser is suddenly going to start spitting out increased damage and adding chunks of DPS, they also don't mean it's going to provide a brand new step by step rotation that couldn't have been done on paper prior to the fight.
What they do mean however is that a person parsing their performance in live content can look at other examples to get an idea of how well they should be performing in a fight and contrary to what you may believe there are many players out there that simply do not know this.
Your example with the two Dragoons doesn't really make any sense in the context of this conversation.
Saying "well this Dragoon did 1300 DPS why didn't you!!!!??" is not the same as someone asking "Can someone tell me what the average DPS of a DRG in this fight is around my ilvl and stats? I'd like to know how much i should be doing." this is where someone can respond with "Well the average burst assuming you're doing your rotation right is usually around 1300-1400 DPS in the first phase give or take with a sustained afterwards of around 1000-1200-ish for the rest of the fight but various factors can change that obviously" this gives that Dragoon something to shoot for when evaluating their own performance.
If That DRG looks at the parser and sees they're only doing 800 burst at the start and start dropping down to 600 sustained they obviously know they are doing something incorrectly but they may have to look at and evaluate their performance to see where they went wrong. Maybe they weren't mechanically fast enough? Maybe they need to adjust their positioning and the way they avoid boss telegraphs in order to maximize their DPS up-time which should be common sense but you'd be surprised how many players don't realize how much stepping away from the boss for an extra GCD or not dodging properly can impact their DPS, they may think it's just 20 points here and there when in reality you could end up dropping a few hundred points instead, which doesn't sound like much until it keeps happening over time and you wipe at 1%, a parser will show you this when you compare the data from multiple attempts.
As for things like BoTD, GL, etc that yes should have 100% up-time, historically there have been many cases of a party doing either too much or two little DPS causing awkward phase transitions where it falls off. It's not about when you activate it but entirely about where you might have to alter your rotation to extend it, the number of times I've heard "Damn i lost GL, Enochian, BoTD, etc" because of an awkward phase are too numerous to count over two years of content ranging from Ex primals that are notorious for lots of jumps when pushed (see: Leviathan, Ifrit, Titan, Garuda) or bosses that you need to come near full stop on DPS so that you don't wipe the party (Melusine, Nael, Bahamut) a parser doesn't change the math in any way, but it does show you your performance counting the variable that is live play.
Funny that you say this, because we don't rely entirely on the math when building things like spaceships we extensively test and re-test every single part down to the bolt before we conduct a launch and even then there are times when we get it wrong or something happens that we cannot foresee that wasn't taken into account. It could be as simple as human error or something entirely beyond the scope of our predictions, you can look up the space shuttle challenger and see how even though everything in the math was probably correct a simple O-ring malfunction can cause disastrous results that defied their predictions and nobody saw coming, sure we know exactly what happened now, but that's because we were able to asses and analyze the data after the fact.
Additionally while we might not reinvent the wheel, we've surely innovated it about a billion (not an actual number) times to suit our various needs, from the materials we construct it with to altering it's form, changing things from the draft board in response to live feedback is exactly how we make progress.



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