Horizons is THAT show.
Ash is a Champion and as far as we are aware has travelled through more regions than Paul has so it’s different. And you are right with this statement “Paul will always jump on the chance to get stronger and be more competitive, as long as he can do it without being burdened” I just think this applies to almost every other person in the Master Class (they wouldn’t be there if that wasn’t the case). It virtually applies to everyone even beyond the Master Class. Most people in the PWC are competitive and want to have opportunities of getting stronger. Also when I made that statement I wasn't excluding Paul but instead I was just saying that the specific reason you mentioned isn't an exclusive reality to only Paul and therefor should not guarantee him a spot.It might not be a good excuse for you, but it is what it is. If DP has showed us anything, Paul will always jump on the chance to get stronger and be more competitive, as long as he can do it without being burdened (though this may change). DP went to great lengths to showed how Ash and Paul are essentially two faces of the same coin. It's pretty ironic you can make the bolded statement but somehow think Paul is the exception to it.
And the Gym + traveling around argument can be applied to Ash as well, which as you know, he's not doing in this series. Besides, nothing about Paul stated that he only needs to travel around. He only travels to wherever he can take the next big step, otherwise (as shown in the Veilstone episodes) he can retreat to a home base to take a breather and take the next step where he needs to go.
I also made the bolded statement because I feel like people took the “they’re images of each other!” and took that to a greater extreme. It’s ignoring the fact that it’s possible having similarities but not doing everything another person does. This is the same type of narrative being used to justify that he must be strong just because Ash is at the moment. Sasuke and Naruto were displayed as being two sides of the same coin and implied multiple times throughout the series but they ended in different places despite that being true. Ash competing in the PWC, being Champion tier and being in the Master Class doesn't mean Paul should be or is even doing that.