I wouldn't say "kill off", but I do think it'll significantly reduce their marketshare, either by a significant number of people migrating from the existing services, and/or by Apple significantly expanding the size of the market by getting significant amounts of people who've never paid for a streaming service to now start using Apple's music service.
Speaking from firsthand experience I can tell you that I just cancelled my Pandora subscription. I did this partially due to the hope and expectation that Apple's service is gonna be great, but to a much larger degree I did it because I started using Pandora like 8 years ago when it was the only game in town, but over these past 8 years pandora hasn't improved at all, while all these new services popped up that are a good or better. I realized that I was still paying pandora 5 bucks a month but I was never using pandora. So Apple's announcement of their new service was enough of a stimulus to get me to finally cancel my pandora subscription.
In the meantime I've been using Amazon Prime Music. It has a *very* small selection of music, which is a HUGE negative, but I get to use it for free because my folks pay for prime only for the shipping so I get to use the included digital services for free, and that's a HUGE positive.
But even with Amazon Music being free for me, if Apple's service turns out as good as it looks like it's gonna be then I'm gonna dump my free Amazon Music and start paying for Apple's Music service.