How does the Galar League work?
Personally...
I like your idea that the minor league is open to Trainers without an endorsement. Keeping in mind that endorsements are not exclusively a matter of highly respected figures recognizing talent in select individuals, but can also take the form of corporate sponsorships, this makes sense to me - the business sector of the Galar region has clearly sunk its claws into the major league and transformed it in part into a platform for mass advertisement. So where do the Trainers who can't get the hookup go to participate in some formal battling action? But because of the inherent "weakness" of the minor league in comparison to the major league (with it being composed of the Gym Leaders who lost the qualifying matches against the Top 8), there is not nearly as much glory and prestige awarded by winning the minor league. Though I would imagine that many of the people who do well in the minor league might have an easier time negotiating endorsements for the next year's major league season.
Might be worth noting that Klara/Avery are explicitly training to become a Poison-/Psychic-type Gym Leader which are currently minor league types. Avery in particular provides an interesting detail, since his League Card establishes that every Psychic-type Gym Leader in Galar's history has come from Avery's bloodline, and Avery himself is seen as something of a disappointment by the rest of them because he lacks prowess with his own psychic powers. Meanwhile, Klara is said to have trained at the Poison-type Gym in hopes of becoming a Gym Leader, but dropped out when it proved more difficult than she'd expected. (Her League Card also says that part of the reason why she chose to specialize in Poison types was because there was "less competition." So maybe it's not a very popular type?) All of this leads me to believe that you have to first become a minor league Gym Leader before you can even think about maybe becoming a major league Gym Leader.
Also, I wonder if Dynamax is allowed within the minor league? Do they have huge stadiums too? My instinct says probably not, so no.
I also wonder about the structure of the minor league, since it has to cover 10 types as opposed to 8. Maybe you only have to win 8 out of 10 minor league Badges to participate in whatever the minor league's finals are, kind of like those times in the anime where we see someone with different Badges than the ones that are earned in the games. They must be challenging unseen Gyms that Ash never encounters.
> Also, what happens to Leon now that he is not the Champion? Would he have to take part in the Gym Challenge again?
My gut sense is that someone who has been Champion once cannot hold the title again. Obviously they can still take part in exhibition matches against the Champion, as that's what the post-game tournaments are (as well as the battle between Raihan and Leon in the opening scene), but I like to think that there is a tradition of accepting that your run as Champion was your spot in the history books, no do-overs. But that's just my interpretation - I'm sure one could just as easily imagine a former Champion being able to invoke some special challenge clause against a sitting Champion for the title. I don't think it would make sense, though, for a former Champion to have to go through the Gym Challenge again, because a former Champion is going to be incredibly strong and would steamroll the rest of the competition unless there were strict limitations in place on what Pokémon they could use. But even then, they'd still have scads of experience compared to Trainers who are just starting out... but then again, Gym Challenger Theemin looks quite old, so he probably has a lot of experience, and yet he can still participate in the annual challenge.
(Incidentally, I would love to have a second round of Galar games that focus on the minor league. It is a far more intriguing prospect to me than any of the remake options that are on the table.)