This is due to a little weirdness that happens when you increase the effective range of a weapon with amplitude and frequency.
The amplitude thing doesn’t really matter, it’s the frequency that’s important here.
To simplify it (because I’m not even that good at this type of math lmao), frequency essentially means how many times it does a sine wave pattern:
Here, the frequency is 1.
In realm, this effectively translates that, within a weapon’s range, it will try to make that movement.
As an example, take the Staff of Esben: it has a frequency of 0.5, meaning it will only make the first half of that wave. And because it has two shots, it creates that shot pattern of two shots diverging at first, and converging at the end of their lifetime:
Tiered staves act the same, but with slightly more within their range - a frequency of 2, so it converges 4 times.
Now, what happens when a staff gets affected by Inspired? This means that, because its range is longer, the effects of its frequency will compensate for it.
It’s basically like taking the previous image of a wave, and then streeeeeeeeeetching it out:
The length between the points where it converges is now much longer than before! That’s why their shot pattern changes.
Back on topic, Inspired is indeed a 1.25x speed increase. This means it cannot affect projectiles which have 0 speed or which move independently from speed (f.e. the first second of a Void Blade shot, and any parametric shots like Bulwark, which only use lifetime and magnitude to calculate their speed).
The reason it isn’t lifetime, which would benefit those weapons, is that a lifetime increase isn’t much more helpful other than increasing range. Speed, on the other hand, means that you can also let shots hit faster things - greatly helping with items such as the
Garfield Garland Bow or Leaf Bow. Of course, it may frick up your normal, slightly predictive aiming, but that’s only a minor nitpick :P