Erinyes is actually a great dogfighter. If you go on multi, you'd realize how dangerous those things are. However, it might not be as suitable as it should be in campaigns because the player is usually in a furball rather than a duel.
The Ulysses is also great at dogfighting. Profile matters a lot. Here's something else that matters: mass. A Ulysses has a tactical advantage over a Perseus by ramming a Perseus. The Ulysses is actually HEAVIER than the Perseus. When the ram happens, the Perseus pilot would be disoriented from two things: The unexpectedness of the ram and the unanticipated direction change that follows. The Ulysses, being more maneuverable, smaller, and having more mass, is more likely to destroy the Perseus. That, and in can spin circles around the Perseus.
However, I still prefer the Perseus because I can outmaneuver an Erinyes and outdamage a Ulysses. The Perseus is my middle ground for the two fighters. I sometimes use the Erinyes but almost never use the Ulysses. Mostly because Kyad uses it. :<
EDIT: There was no mention of heaver.
My opinion on the Erinyes: In one-on-ones, the Erinyes can actually stay still while the opponent has to keep maneuvering to avoid getting shot. The Erinyes has little need to move other than to make the opponent adjust more. Even if the Erinyes gets shot a first by a second, the damage output by the Erinyes would quickly vaporize the more maneuverable fighter.
However, if you add more fighters, teams included, the Erinyes cannot stay still. This limits the ability of the Erinyes to pressure the opponent because if the Erinyes does stay still for a long period of time, someone else will gank it.
Also, the more bandits there are on the field, the less able the Erinyes becomes. Being energy-dependent, it would have to readjust its ETS every time. The Erinyes depends on all three to actually be useful. If it has no shields, then the fighter would break because it's huge. If it has no engines, it cannot avoid knife fights with multiple opponents nor can it avoid missiles. If it does not have weapons, it will not shoot at all.