So do mine. So do many DMs' NPCs, I'd wager, so this isn't very unique in my experience. It's pretty easy to come up with those motivations, relationships, and threat assessments. Even on the fly when they've turned their sights on the familiar. I'm still not sure this gets to the heart of the objection though.
The phrase "kill on sight," in general, means that the killer drops whatever they are doing and focuses on killing the target. Literally, as soon as one can see the target, one attempts to kill them, basically no matter what. To engage in "kill on sight" means to put killing the target of that attitude at the top of one's priority list, and is used to indicate that negotiation or non-lethal options are completely off the table. "Kill on sight" is just about
maximally strident, I can't think of a more actively hostile stance--it comes across as being willing to drop pretty much every other priority in order to destroy the familiar as soon as one is aware of it.
"If given the opportunity" does not ratchet this response down to something moderate. Frankly, as far as I'm concerned, the phrase adds no meaning. If one has no opportunities to attack, then whether one's attitude is "kill on sight" or not is irrelevant. Sort of like that old joke about the object that repels tigers or the like--"but tigers don't live around here." "Then it's WORKING!" "If given the opportunity" just means...it's not
impossible to try. Maybe--if I was feeling particularly charitable--I could allow something like "I won't take extreme risks to do it, but I will gladly take moderate risks."
As far as I'm concerned, the familiar literally being
present for combat IS the opportunity to kill it. Like when the Eleventh Doctor used the famous Apollo 11 landing broadcast to defeat the Silence ("This is one small step for man..." [bzzt] "YOU SHOULD KILL US ALL ON
SIGHT" [bzzt] "...one giant leap for mankind.") Every human that watches that broadcast and then sees a Silent later drops what they're doing to kill said Silent, and then (because of their innate powers) forgets they've done so. Adding "if given the opportunity" does not lessen the "ah, a familiar, I
must kill it
immediately" meaning.
Now, if you'd said something like, "My general policy is treat familiars as fair targets in combat regardless, especially if they take risks or put themselves in melee range, and I let the players know that," that would have had rather a different impact. As it stands, yeah, if you would call your stance "kill on sight," I very much think that means "I will prioritize killing familiars over other actions, should it seem possible (or at least remotely feasible) to do."
Edit: Yeah, as Lyxen said. "Kill on sight" means one
creates opportunities, takes meaningful risk to try to secure the kill. Perhaps not ridiculous "I'll gladly risk total defeat just to try to kill the familiar," but still willing to delay or forestall almost any other goal simply to try to kill it. "Kill on sight" means the moment you see it, you want it dead, period.