I think that is very unlikely that Emery even had 1% influence on Purslow leaving. In the end he is only the head coach (not even manager - clubs own words).
I do think that when it comes to sporting decisions like signings, his words will hold a lot of value - although he probably is not making the final decision. However, I do not believe that Emery has the power of getting Lange fired or hiring Monchi for example. He will of course be heard in the Monchi recruitment process, but in the end Monchi will be the boss of Emery, so it does not make sense to say, that Monchi is hired as part of Emery's team.
I also hope that it is this way. No matter how positive everything is right now, we are coming out of a season where half of the clubs in PL sacked their manager. If we by January are sitting in 13th and out of the cups, it is not unthinkable that we would do the same.
Relying heavily on a single manager and the lack of stability that gives you is how not to run a football club. However, from everything I have read, I am also very comfortable, that NSWE sees things the same way.