Where Jira really shines isn't in moving tickets along a man an board.
It's in supporting the metadata that allows your engineers to quickly refer to a ticket (by number) and your project manager (or whoever serves that function) to search and filter the ticket inventory to find what needs their attention.
I've always worked on small teams. I've tried almost every issue tracker I could get my hands on. Yes, even Trello (or <shudder> asana) is better than nothing. But once we've got more than 2 engineers or more than 50 tickets in our backlog (500-1000 is a pretty normal count for a company that's been around more than a year or two), I invariably find myself wishing desperately for this or that feature of Jira.
The problem with Jira is that it's designed to be customized and locked down for the most rigid and enterprisey of large corporations. The configuration is insanely complex and has a ridiculous learning curve. And the amount of process it introduces in its default configuration is indeed way too much for a small team.
But if you're willing to spend a day or two adjusting it, you can set up a Trello-like workflow while still gaining features like advanced ticket searching, automatically reassigning tickets on status change, release tracking, web hooks for CI and version control, multiple views of the same data, bulk operations, man an board swim lanes, and other conveniences that will make your PM and devops people more efficient and less frustrated.
Those things aren't distractions from your users' pain points. They're features to address your own team's pain points so that they can focus on the users' pain points - instead of spending their time wading through poorly organized information and their energy trying to make sense of that information.