I'm very interested to hear @Shaggy's take on this, but in the meantime here are my thoughts...
Culture, industry, geopolitics, economics, and entertainment trends have changed A LOT since the "heyday" of Six Flags Kentucky Kingdom—an era when SIX was truly invested in that park and its potential. A time back when there were talks of big-budget investments that forced then-Paramount's Kings Island to respond in kind (i.e. SFKK's plans for a "Gotham City" that allegedly led to the creation of "Action Zone" up north). Since then, KI flourished and SFKK, well, we know how SIX treated that park. One of the reasons that I've always heard from industry folks about why SIX backed off so much from KK was that the park always had a strong season pass base. Essentially, even if the new additions were paltry compared to other similarly sized parks, the season pass holders kept coming. A strategy that might work for a short time, but certainly doesn't work in the long term (especially if you keep cutting and are under the corporate umbrella of a debt-ridden Six Flags corporation). At the same time, (P)KI has also always relied on a very strong season pass base, but also attracted (and presumably still does) a good amount of travelers, particularly regional travelers.
So, all that being said, is there a "threat" (not implying that this is Baconator's word, but I mean this from a competition standpoint) from a rejuvenated Kentucky Kingdom? That depends on how many people in the Louisville metro/market/region currently make a trip or two to KI and spend money at that park. Will they now be wooed away by Herschend's (or previously Ed Hart 2.0's) Kentucky Kingdom? Hard to say and even if they are... does that even make a difference to KI?
Back in 2006 (so, yeah, this is an old example and a one-off interaction), I had a conversation with a family from Louisville who was visiting Kings Island for the day. This trip was one of their vacations that summer. They had a great time, but were disappointed that Kings Island's Action Theatre was showing the same Spongebob movie Six Flags Kentucky Kingdom was showing (they had waited in line for a decent amount of time). They wanted to make the most of their day because this trip was a real treat for them. My point in mentioning this is that family, despite being from Louisville and having Six Flags in their "backyard," viewed (Paramount's) Kings Island as a superior experience or at least a destination worth going to even with a theme park of their own.
I'm not sure how much, if at all, Kings Island advertises in the Louisville market, but I do think it's interesting that Kentucky Kingdom (ever since its rebirth) has advertised in the Cincinnati market. When I paid a visit in 2017, they even had a great deal specifically for "out of state" visitors that was clearly geared at this market (billboards all over Cincy had advertised the deal). I don't think the two parks will ever compete for season pass bases (their bread and butter), aside from maybe a handful of guests that live somewhere in the middle. But they may (especially depending on how Herschend develops the park) compete for regional tourism dollars. But to that point, the industry (and entertainment in general (and entertainment post-covid)) has changed a great deal and the two parks themselves are very different from the PKI and SFKK movie theme park days. Kings Island is certainly a much bigger heavy hitter compared to the two, but the distance doesn't make this a PKI vs Americana situation.
Herschend is probably more concerned (I assume) with establishing and growing a loyal season pass base and if they do feel competition, it's from a park in Santa Claus. No doubt both KI and KK currently do (and will continue to) keep an eye on each other's movements, but I'd say until (or if) KK becomes some sort of true destination, there's going to be peaceful coexistence.
In regards to Herschend becoming a "heavy hitter," I'm very excited to see what they do with the park. I've never personally been to Dollywood, but that park's reputation certainly speaks for itself. It's an industry darling and its attractions, events, and staff are well regarded. Louisville is an interesting market (a large metro, even if it's not a "major league" city) and the park has great potential to be a true crown jewel of the area's tourism/local pride.
The first time I ever visited Kentucky Kingdom, it was shortly after Six Flags had abandoned the property. Ed Hart's group was pushing for a new plan (before Bluegrass Boardwalk was a thing) and I was able to tour the property. I assumed there was absolutely no way that park was ever going to reopen given the state it was in (some sections had been closed even longer), the ridiculous land and lease situation, and trying to find a viable group. After the Boardwalk fell through, I just assumed the park was done. I was certain of it. Lo and behold—I finally got to visit it in 2017. I was so happy to have been so wrong. What a great park that's really doing some great things these last few seasons.