Thunderbird is great. But if you don’t like it only because of Mozilla (the interface is customizable so that’s not an argument) there is a fork which does almost everything like Thunderbird, is slightly better on some points, and slightly worse on some others, called Betterbird. It has in my point of view no interesting point, but it’s not Mozilla.
Which version do you use with which OS? As a GNOME user I don’t need it, but I’m pretty sure Thunderbird has a system tray.