TL;DR: Ships kinda suck now, but adding more ships and possibly new mechanics could make the mode less stale/repetitive and more fun.
Look, most of us agree that ships is a mess, and there are few common complaints: it's too RNG dependent, and the 3v3 format only exacerbates the RNG-fest. So I'm not going to harp on that all over again. Instead I'm going to offer some constructive suggestions that might make ships less boring and more fun.
IDEA #1: Add a lot more ships
Fleet Arena has always been fairly stale. And going through the ships PvE table got boring quickly because I was facing the same ships over and over again. Really, one of the biggest problems is that there are just too few ships.
I don't have a list of all the new characters and ships that have been added to game, but it feels to me like we get maybe five to ten new characters for every one ship that gets added. Why does ships feel like the neglected, half-***ed part of the game? Because it sort of has been.
Adding new ships and their kits could also be a simple way to correct or address some existing problems, such as RNG determining a battle in the first turns.
IDEA #2: Spice up the mechanics (for example, with bombers)
Ships, in my opinion, has boiled down to a few rather repetitive mechanics: Target Lock and Taunt (which converge in the ubiquitous Biggs).
What if bombers were introduced to ships? This could add another strategic layer to the battle. Bombers could be able to target capital ships. Since you can't kill a capital ship, bombers could apply detrimental effects, such as daze, ability block, cooldown increase, offense down, or removing TM.
Fleet battles could then evolve into not just back-and-forth slogs of taunt vs. target lock, but a game of trying to keep your bombers alive to attack the capital ship while using your other fighters to prevent the enemy bombers from getting to yours. (And maybe a return to 5v5 could accommodate the extra strategy involved?)
Additional tags could allow for more strategy: for example, interceptors could have bonuses against bombers.
Addressing obstacles to adding more ships
One major obstacle to just adding more ships, which I'm sure you've run up against, is that ships have to be tied to specific characters. Ships can't just be added willy-nilly, but they have to wait until a pilot character is added to the game. This can be a real challenge to beefing up the ships roster.
But first off, there is already some low-hanging fruit in the game: JKA could get a Jedi Starfighter. Bossk could get Hound's Tooth; Dengar, Punishing One; IG-88, IG-2000. (And I know you're keeping a Han and Chewie version of the Millennium Falcon up your sleeves, and that's fine.)
At some point, KOTOR characters could pilot the Ebon Hawk, and the players would love it.
However, there is also room to add lesser-known characters for needed roles. As an example, Colonel Stark was a complete nobody character who was added to meet a specific need. But his kit was good enough that his obscurity didn't prevent players from adopting him eagerly. Lesser-known characters can be well-received.
So there could be room for, for example, Jon "Dutch" Vander or Horton Salm (Y-wings), Tycho Celchu or Arvel Crynyd (A-wings), Major Rhymus or Captain Jonus (TIE Bombers), Soontir Fel (TIE Interceptor), Maarek Steele (TIE Defender?). Seriously, look at the old SWCCG or the X-wing Miniatures game.
Pilots abound in Star Wars. And we seriously need more ships to keep that game mode interesting.