I've been in this holy-wars among with other agile enthusiasts hundred of times, and this is my final executive opinion:
There is no way to fix organizational mess with the tool, no matter how good it is.
Sad, but true. Usual pitfalls such as messy backlog, unsuitable workflow that confuses the team, lost communication between teammates are not about the tool, they're about how the teamwork is organized. That is why my advice is to pay attention to team environment design (which have to include Jira-ways-of-working) all the time the development goes.