In my opinion, there are a few distinct routes the series could go down.
1. Rail Shooter
This is the series norm, where most of the gameplay is moving down a set path with limited movement within said path. This is the method used by Star Fox, Star Fox 64, Star Fox Zero, etc.
Pros: This has historically been the most successful type of Star Fox game. This would likely be a relatively straightforward type of game to make, since you wouldn't have to reinvent any wheels, just stick to a formula that's had decades of polish.
Cons: Rail Shooters don't actually sell all that well. They are also hard games to make a lot of content for; you can play every possible route in Star Fox 64 in a few playthroughs totaling maybe 10 hours, and even Star Fox Zero can be played through in a long sitting. Also, how many devs out there are skilled at making big production rail shooters nowadays.
2. Ace Combat-like
This type of game would take cues from Flight Action games like Ace Combat. Though not exactly flight sims, these sorts of games would require more complex controls and settings.
Pros: Ace Combat style games can be successful. These games have plenty of content. Namco has worked with Nintendo plenty in recent times and can do so again.
Cons: Flight Action is less easily approachable for newcomers than Rail Shooter gameplay. This would arguably be a drastic divergence for the series.
3. Variety Third Person Shooter
What if some levels were rail shooting segments, some levels were on-foot segments, some levels had you fighting in a Landmaster in urban areas, etc? Star Fox Assault did something like this.
Pros: Easy gameplay variety. Probably easiest to make multiplayer for.
Cons: Possibly harder to balance and polish due to multiple gameplay styles. Arguably what fans would be least interested in.
