I think she needs an even bigger change: let tentacles killed within X range of the vessel count no matter who kills them.
It's not a matter of power level or distribution of her power between different abilities (i.e. if they feel they need to offset this with a buff somewhere else that's fine with me), it's about what kind of play is rewarded and punished. If Illaoi is my enemy, I should be rewarded for killing her stuff, not have my teammate start getting pissed off at me for "ksing" a tentacle. Conversely if I am the vessel, I shouldn't be stuck with longer vessel status because my teammates are trying to help me and keep Illaoi from getting too overwhelming a tentacle nest.
League is a team game and having your teammates help you with your problems is how it is supposed to work. Illaoi should be able to appreciate that in lore, IMO, but even if she can't, the actual gameplay of League should be pro-teamwork. Even if Illaoi personally disapproves of having your friends solve your problems, she shouldn't be able to stop it from working.
People SHOULD have the habit of whacking the tentacles when Illaoi is their enemy. They're enemy tentacles! The problem is not that people are hitting the tentacles, it's that hitting enemy tentacles is sometimes obscurely counterproductive.
Better clarity for vessels would probably be good too. But IMO that's a side issue compared to the anti-teamwork mechanic.