I think some type of hardened objective assault/defense (like capital ship) is what would garner most popular support and hype.
And it shouldn't be too hard to implement in the current GSF engine, as long as the objective doesn't move.
Basically, just make a big capital ship, put it in interesting terrain that blocks long range LOS to some elements (like a huge repair facility), and slap a very big number of very durable targets/turrets on it--each with different defensive qualities that reward different weapon choices, ship classes, and components.
This. Most definitely.
One particular advantage to such a mode is that it would make bombers actually act as bombers. Load up protorps, go bomb the enemy! Likewise, Strike fighters would play a greater role.
In fact I believe there was some datamined info that hinted at a game mode like that. Hopefully that mode wasn't scrapped, but merely put on the back burner.
Regardless, I think the single most important update to GSF would be a new game mode to add actual variety to the gameplay.