Not to the degree you think. Let me show you two examples:
Look at the Cardinals. BA and Keim have an understanding about staying in their own lanes. They've been asked on multiple occasions about that dynamic, and it's a mutual understanding about how Keim brings in the guys who will fit Arians's scheme. I think the only move that BA lobbied for heavily was to sign Drew Stanton. The team is looking at three double-digit win seasons in a row.
Now, look at the Eagles. Chip Kelly has taken over the reigns as coach and the person who has final say in player aquisitions and departures. He rids of a productive running back in McCoy in favor of getting Murray, who is struggling behind the garbage o-line. Kelly makes an unnecessary trade for Bradford, who's been marginal at best and is an injury waiting to happen. Kelly has basically overturned Reid's entire roster with only marginal production to speak of. Yes, he won 20 games his first two seasons, but as soon as he took over in the player/personnel department, the team has looked incredibly mediocre.
When coaches influence rosters, more often than not, it's a disaster. When coaches coach, and GMs GM, and there's an effective working relationship and mutual understanding between them, that's the ideal situation for any team.
Whiz's biggest mistake is not with coaching, it's with his own stubbornness and not understanding where his primary responsibilities lie. Being a head coach is a lot of work by itself; taking over GM responsibilities is career suicide, which Whiz's has committed twice. This does not make him a bad coach by any stretch of the imagination, it's about him doing more than his job title entails.