I've used Jira, spreadsheets, Trello, Asana, and YouTrack. All of them are capable of running large teams. All of them have needs in order to use them effectively. Here they are:
Jira, Asana, and YouTrack are all highly structured, and frustrating to set up workflow tools. The forced structure is useful when having to do the minimal work necessary to keep users from messing up the flow. It takes a lot of time to set up the way you would like it, and it causes a team to conform to the way it functions. For some, this is perfect. It tells you how to organize. It forces users to do it the way it wants. It's cumbersome to change it. It's constraining.
Trello is a blank canvas. It provides you the tools to create an organized work flow. It's flexible, it highly graphical, it's easily broken. While I prefer Trello due to its lack of constraints, it's also its Achilles' heel. Without strong organization, and persistent upkeep, Trello can very quickly go awry. This becomes more difficult as it scales, and more users can use it incorrectly. This requires team commitment, and a person designated to maintain the boards. When its working, its wonderful. When bad management sneaks into it, its horrible. '
Choose wisely.