I absolutely believe that no paid employee of Travian should play on any server. That is a clear cut case of conflict of interest.
Chat Moderators and Support team members are volunteers, not paid employees. They are giving of their own time to help out with a game they enjoy playing, and continuing to play while "working" not only makes it worth their time to log in and be present but also keeps them up-to-date on how any recent game changes affect the game play and other potential issues experienced in the game.
Now, let's skip the Chat Moderator side of things, because they don't actually handle reports. So any conflict of interest that could arise from how reports are handled would land solely on Support.
There is NO CONFLICT OF INTEREST unless a Supporter is dealing with something which affects themselves, such as reports generated from a server they actively play. Every report submitted is reviewed by the lead CSR, regardless of who initially responds and/or who resolves it. As Samisu noted, every report which results in any action by Support is very clearly documented, and the Supporter can be called on to give clarification on things when necessary.
Samisu made a comment about how team members can find themselves targeted by players, and I can say it definitely happened to me. One of the reasons I stopped being a chat mod was because I had a player follow me (literally choosing to join whatever city I played on the map, bringing friends after the first time) through 3 rounds so he could trash my home city and throw verbal garbage at me every chance he got. It wasn't the only reason, nor the only time I'd been a target, but it was one of the last straws. I'd have quit a lot sooner, though, if I'd been banned from playing just because I wanted to help out.