I doubt switching the bugtracker to a different one would help. The issue is likely to be at some lower level, mail server misconfiguration or something. We don't need to replace the tools, we need people in the sysadmin team to take care of the servers configuration and keeping things running. If they can't do it, we should consider migrating to a hosted platform where someone else does the work for us, but then the work to migrate our existing ticket base… also requires someone to invest effort in it.
I think overall we are quite happy with Trac the way it's set up. So there is no need to replace it with another tool. We just need to get the server configured properly, and only a few people have ssh access there (security reasons, plus you don't want everyone to mess with the setup).