The self-insert thing is always just that dumb thing you do when you're young, and occasionally when you're not, but it's still a mistake. I'll admit that one character I've been interested in for too long was borne of a self-insert and evolved into... Not being a self-insert.

Amnesia-now that is a tricky, tricky sunnuvagun to work with, and deceptively so. While the plot with Kane and Anthony doesn't seem too bad in and of itself, it's rather pointless to deal with total amnesia as Anthony has as a massive plot point. It really only works if there's a lot going on behind that amnesia, and simply put, there hasn't been. Anthony's day-one goal has been to retrieve his Blastoise and take down O.X, and absolutely nothing thus far has implied his amnesia is doing much more than complicating matters and thus making a hassle in the plot. (If Anthony were revealed to formerly be O.X. then that'd be a completely different can of worms, but that would likely have been brought up earlier-"Traitor!" Plus Kane would likely have, in effect, turned into an expy of Silver in that he hates the evil team and fights against them, but has bad blood with the protagonist, as no way would Kane have joined up in the same faction as anthony willingly.) It's much more useful to have major assumptions about the plot outright upheave as a result of that amnesia. (Or, as Planescape: Torment teaches, have the quest for identity be the primary motivator and plot in and of itself). It would be much more of a payoff it it were revealed, say, Anthony was a Legendary Pokemon that winds up being relevant to the plot, and his 'memories' are copies of the current Kanto Champion as a means to keep his cover consistent and powerful. (But that's even less likely than the O.X. example above, plotwise.)

Given the way the plot is going, I can meet with you on the whole issue with Badges, but there is a way to work around the tedium of it. They're called Pokemon Gyms, and the way things are now, Anthony is, so to speak, out of shape. Having a Gym Leader be personally involved with the (re)training would make an interesting amount of sense, beyond just wandering around, and not only make the Gym Leaders relevant to the plot in terms of supporting Anthony (possibly more intimately than just as retrainers), but also really prove his worth and make inevitably getting that badge turn from an 'eh' milestone to something that basically vindicates Anthony's progress as a trainer and proves he's ready to take on Organization X again. Even the general formula makes sense for anthony as a form of-retraining himself. After all-three years of aimless wandering and fighting Organization X in skirmishes have done zilch for him.

I honestly don't care too much about the chapters personally, but it does make it feel like every step... Yeah I completely agree, or at least I see why the OCD alarms are triggering.

Eevees have always been handouts in canon Pokemon as well, but really, it's more like Eevee is a go-to Pokemon for this kind of stuff to the point of cliche, especially showing off Pokemon bonds and such if they evo into Sylveon/Umbreon/Espeon. In the future, you can prove that same point in a somewhat more low-key/non-sue manner, like Crobat (requires happiness to evo like the aforementioned trio of eeveelutions and on top of that is excessively common, meaning a lot of trainers could have Golbat but very few could have Crobat), Mega Evolution/Z-Moves (require a strong bond between Pokemon to use in-universe), or excessively powerful Returns (grow in power if the user likes their trainer). I mention this because Anthony has an Umbreon and two of the girls have Sylveon/Espeon, so different ways of showing that off could be useful to you.

Keep on chugging, man. I can tell you that the quality of your work and writing has improved! I'll continue reading if you continue posting.