You can in GTA5, but I have not touched that game in months. That sandbox is not designed to be played (long term) without a story. You can drive/fly around, shoot things and get chased by cops, but that's pretty much it. There is the GTA5 online side of things, but that's a rats nest of random human interaction.Consider for example if Mass Effect allowed you to explore the entire galaxy once you finished the story.
Elite was designed to be dynamic. It is not at it's potential yet by a long shot, but there's always something to do. The layers being added now and in the future sit on that foundation. A single player cannot affect the galaxy in any appreciable way and that frustrates many.
Any mission system that complies to a story or plot will be finite by nature. Once all other things are in place, its possible Frontier might be able to drop in linear missions, but they would have to be Solo mode only for obvious reasons and that kills the MMO portion of the game.
There is no perfect solution that will make everyone happy, but keeping Elite dynamic is working to engage players presently. Yes, it needs to be a lot more dynamic, but I sense that is coming. Given the amount of work that needs to go into AAA games, it won't happen any time soon. We just need to be patient and engage with the devs to (hopefully) stay on the right path together.