Now there's a workable definition. Now looking at BW, we start to see companions being defined by professions that don't involve battling.
Cilian for instant is a
Pokémon Connoisseur, which a profession involving judging a Pokémon's bond with its trainer and how to make it better. No battling is required. Clemont is an inventor who creates a robot Gym Leader just so he can have more time to do more inventions rather than do Gym Battles all day. Invention is his passion. Bonnie is well too young to do Pokémon training. XY in total, none except Ash are interested in competitive battling.
In SM, we see Kiawe is obsessed with battling, but the others vary in terms of goal. Mallow is related to cooking, Lillie is more of getting over her fear of Pokémon and fixing family issues, and Sophocles is more interested in inventions and Charjabug racing. Lana is more vague when it comes to goals, but it's generally considered to be mastering fishing and exploring for new Water-type Pokémon. A goal doesn't technically need battling but considering the circumstances of wild Pokémon attacking, it is recommended just like Goh facing off Pokémon who are not inclined to be so caught easily.
Which is why I point out that Goh usually throws a Poke Ball at a Pokémon without ever needing to battle at the beginning. Battling may be an option, but it's not a technically requirement. While Project Mew may have trials that require battling, such as raids, is it any different from Totem Battles in Alola?
As we see with the supporting cast, very few of Ash's supporting cast (which is really just Kiawe) have competitive battling as the core aspect of the goal. Even if we push Goh into that spot alongside Kiawe, it's still not that much. All it really shows is that battling in general went down, being distilled to just Ash. And considering that SM is more about slice of life than Pokémon battles, I wouldn't be surprised if Ash battled less frequently in that series as well. When Journeys came around, Ash became full-time battler and he got himself a lot of female rivals that exceed any League tournament he faced, with Bea being a true rival to him (and the first recurring female rival to boot). And really, Ash truly monopolize the battling aspect given that Goh's goal and Project Mew has more of a variety that isn't just about battling. In fact, the whole Mew conflict is resolved not with a fight, but with words.
Looking at this overall, it's not really that simple as to say the anime wants no girls being battle competitors. Rather, the series was distilling everything relating to battling to Ash's story, with folks like Gladion and Kiawe becoming rivals to his story. Which is I believe Ash has been the reason for this appearance of decline. The games abandoned contests, but Leagues still remain as a stable of the franchise aside from "Gotta Catch 'Em All." As long as Ash remained the main protagonist and the anime distilled Pokémon to those basic concepts, it makes sense as to why Ash had to go in order for a more battle competitive girl to debut and thrive.