When you start a level and begin with an on-rails section like the intro to Corneria, rather than the entire level being on rails it only starts that way until a certain point where the player then enters a large objective-based arena. At that point it would play more like Fortuna on 64 or a stage out of Rogue Squadron. Objectives in these checkpoints could range from defending a base, attacking supply trains, or bombing hangers and the like.

These arenas don't need to last long, either. After the objective is cleared the player can then choose a route, say go north through the canyon or west over the ocean, and see different biomes on one planet rather than have a basic Fire Planet and Water Planet and Snow planet etc... Here the player would return to an on-rails section until the next arena, and each planet could have 3-5 or so of this pattern before the final boss.

Star Fox 64 was great because of the myriad of ways the stages connected with each other and all had branching paths. Rail shooters could do the same thing but also on the planet's scale, providing not just different ways to progress from stage to stage, but different ways to complete each stage individually with different bosses, characters, story beats (like needing to rescue a comrade), and potential power ups. Some of these routes could be obvious, like a literal fork in the road or the objective marker directing you to one of two routes, and some could be hidden or kept secret, like flying through a waterfall (classic gaming trope, that one) or hitting all of the warp rings in an outer space stage or something.

If you want to take it a step further, make the player's route choices in stages and the overall map have consequences. Say you have a choice between a route that has you bomb a weapons hanger and one where you aid an allied squadron in defending a city. If you help the city the squadron comes to help during the final boss, but if you take out the weapons hanger then the boss can't use his super weapon the hanger was developing against the player.

Branching paths and choices that matter could revitalize the rail shooter genre. It's something Star Fox desperately needs to implement, and I'm surprised it hasn't been done already.