My counter argument is that you can see a distinction in several of these lines between when the characters are gassing up their own ability (Chuck, Will, Bruno, Falkner, and even Clair ironically enough) vs specifically trying to undermine yours in the pre-battle boasting (Lt. Surge, Blue, and Karen as some are discussing). The latter are in 2 cases coming from characters who are generally depicted as arrogant and not particularly approachable/polite in most multimedia depictions of the that timeframe in-universe, be it them or counterparts like Blue-Gary (and Blue still acknowledges you as a Champ and basically says "maybe the Gyms suck, but I'll let you speak for yourself in battle" which suggests he's trying to push your buttons to get a harder fight). Karen talking a big game before the battle is in character, but she's specifically talking in a way that brushes off your ability despite getting to where you are.
"My Pokemon are so tough. Think you can take me on? Just try to entertain me" in most other contexts is used to denote a character who assumes they'll win for sure and is mostly hoping they have to put in an amusing amount of effort (you see a lot of this with arrogant Shonen villains like Frieza in Dragon Ball, or even just joke memes like UI Shaggy). Karen's line would make much more sense to me coming from someone like Bruno, whose pre-battle quote is still very much a "this is how I view strength, show me how you do" kind of spiel.
I think the part I take umbrage with is some of the word choices muddying the sentiment. Pokemon being "strong or weak" is not just down to perception, even within context there is a clear discrepancy between the ability of several Pokemon to battle, such that many clearly are better at avoiding conflict or alluded to multiple times in the dex as fighting dirty or hiding/being weak from informational sources. Some Pokemon simply are not meant for battling by depiction, so it doesn't make one a skilled trainer to try pushing them into something they're not suited for rather than playing to what they are capable of. This also can depend retroactively on if "Trainer" refers specifically to just anyone who partners with Pokemon or specifically Battlers (as opposed to Coordinators, Breeders, people Employed with them, Performers, etc.), though I won't harp on this too much for Gen 2.
Besides that, "truly skilled" is another word I don't like, because as many a strawman takedown will show, it gives this impression that using the best performing Pokemon instead of a mish-mash of Pokemon you like, ability or synergy be damned, takes less skill as a trainer. It's a subject you can't escape when looking at competitive where stuff like CHALK or Restricted Centralization happens in VGC, or Smogon OU is centered heavily around whatever Ground or Ghost mon of a given gen. Trying to use your favorites doesn't indicate skill unless you can take them far competing with others, and if you have multiple favorites, what happens if they don't compete well together; A lot of my favorites include stuff like Toxicroak, Chandelure, Tinkaton, and Bisharp/Kingambit. Is it skillful to use a bunch of them knowing I'll crumple to basically anything with Earthquake? And does this mean I'm more skilled for finding teammates to help my otherwise troubled favorites, or that I should be trying to make this disparate-battling team of Mons I like work anyway?
I'd harp on this a lot less if the word had been something like "dedicated" or "devoted" for the trainer, as it is a better reflection of trainers who try to make a few favorite mons work as compared to those who try to find the most effective battle strategies first. Suggesting it's a matter of skill rings a bit hollow coming from main-game as well because putting aside my gripes about difficulty or AI, you can count the battles that you can't just Unga-through with neutral/SE moves on one hand in any given game (Misty's Starmie, Whitney's infamous Miltank or Morty's Gengar, Lenora's Watchog, Totem Mimikyu, etc), so this isn't a context where using your favorites over the "best" stuff takes significantly more skill or effort unless the player is totally unfamiliar with Pokemon's basic mechanics.