"Exotic Beasts must always end their activation within 3" of their owner. If the Exotic Beast is more than 3" away at the end of its activation, it must pass a Nerve test or become Broken."
It's the word "must". Can you knowingly push the pet out of leash range indefinitely provided they pass their nerve/cool checks?
Yes, because it the first sentence does not specifically restrict movement within a 3" radius, it forshadows an effect that occurs if the "must" condition is not met - which could easily happen if, say, they charge a Goliath with a spudjacker who counterattacks and Knockbacks them over a railing!
The Group Activation [page 101, Core Rulebook] and Exotic Beast [page 87, ibid] rules do not specify that the Owner must move first.
Funny that the Exotic Beast rules appear before the activation rules - part of the problem of writing these books with so much granular detail.
IMO a better phrasing would have been "Exotic Beasts that do not end their activation within 3" of their owner must pass a Nerve test or become Broken".
One sentence, doesn't need further clarification, makes it clear that there's a bubble within which the EB must function lest they get scared and that they aren't meant to be free-roaming extra bodies on the board.
The benefit of moving EBs first: they can charge an enemy fighter and then grant an assist to other fighters subsequently attacking the same enemy fighter if they survive the charged fighter counterattack. EBs such as the Grapplehawk, which has an attack with the Grapple trait (granting the benefit of having rolled a 6 on both Disarm and Entangle when it hits, meaning counterattacks from the charged fighter are unarmed at -2 to hit for the remainder of the round) are obviously intended to be sent in first to then grant assists (alongside the melee weapon shutdown effects mentioned) to a charging owner.
This downside of EBs moving first: they might break at the end of their activation and suddenly the owner has to re-consider their charge. Though the Grapplehawk's disarm and entangle effects from Grapple last until the end of the round RAW, despite the possibility that the 'hawk immediately returns ruffled (and no longer gripping whatever close combat weapon the opponent had...).
The break rule is a bit silly in practice - what is the difference between a lagging-but-within-3" Exotic Beast potentially starting their activation 8" away from their first-moving-master that just 5" charged a a bigger scary opponent vs the Exotic Beast that ends an activation 4" away from their master after being knocked back by a Goliath Spudjacker counterattack? The ability to react rationally.
Weird example - the [House of Faith, page 49] Cawdor Pious EB Sheen Bird's Bate special rule makes it do a charge action towards the nearest enemy fighter when its owner is activated (but before the owner moves) - after that charge action it can then be activated normally (before or after the owner). So it could move 7-9" away from its owner before the owner moves - beginning its actual activation well outside of the Break bubble it must return to. It's the type of trait I'd expect to see on a Wyld Runner's barely-trained Phylnxes (though they have the Reckless trait on their attacks, so the theme is already present there).