Though I as a player have little or no interest in competitive FPS matches, I do think there is a place for them in New Eden. During fanfest CCP introduced their vision for project legion as a game with two distinct pillars.

The new messaging around project legion is that it is a "sandbox shooter." But as the OP has so thoroughly demonstrated, lobby matches with fixed player counts and sandbox gameplay don't really go hand in hand.

I do think that keeping those two styles of gameplay separate and not trying to make one too much like the other is the best approach to providing players with more compelling and satisfying options for how to spend their time in FPS New Eden. Any attempt to collapse the two styles of gameplay could limit player choice, and end up putting more constraints on players than otherwise.

So instead of working on ways to turn competitive FPS gameplay into a sandbox, what are some ways in which you can imagine that sort of think taking place within a sandbox?

The idea of making the lobby match competitive FPS portion of project legion a build your own battle arena has some merit. But with the recent announcement of Unreal Tournament 4 it really looks like Unreal Tournament has pretty much cornered the market on a build your own battle arena competitive FPS sandbox. Allowing players to make their own "maps" in some way in legion is still worth doing. But is that enough?

One of the big draws of Dust 514 was the idea that it could be a blood sport for the entertainment of capsuleers. Sadly, that is another promise of Dust 514 that never materialized. So why not build an FPS sports game, within EVE? That is exactly what competitive FPS games are. When you get right down to it. So why not make something like FPS Football Manager in space? Only, with real players competing for wealth and glory by shooting each other in the face, as they say?

Introduce skills to both EVE and 'project legion' that would allow players to build their own leagues, teams, arenas, gambling establishments. Provide players with the tools to make contracts, host tournaments, pay out prize money, create vanity gear and merchandise. Instead of designing the competitive matches to be about the game, or the maps, or a particular game mode, or about game play, balance, fittings, fairness, why not give players themselves the tools to make a game about all of that and more? Given the right tools, we can make a competitive FPS game of lobby matches (two teams of an equal number of players competing against each other in the same place at the same time) that is about the most important thing of all, a game about the players.

You do realize that the term lobby shooter is an excessively broad term right that it nearly applies to just about every shooter out there. I mean hell minecraft is can be considered one if you think about it.

To the point your suggestions are still only chalking it up to the issue still. Lobbies getting replaced instead of menus with graphics and displayed text instead. It is still a lobby shooter you are suggesting.