The theory isn't that Paultons operate in the market completely independently of Merlin, but that the competitive overlap is small, and that there is little direct "competition" with Merlin parks, especially considering geography and how regional Paultons is. If Chessington closed for good tomorrow, half it's visitors wouldn't suddenly descend upon Paultons.
A reference to queues and Fastracks in marketing material is likely to do with people thinking that these things are customary theme park practice, and therefore detractors from visiting a theme park in the first place. It's not saying "don't visit those Merlin scoundrels this week, come to us instead" anymore than it's saying "visit us and enjoy short queues that you may otherwise expect us to be charging you for". Not that a social media post can be used to define an entire businesses marketing strategy anyway.
The same is true if Alton Towers was flogged off. A new owner could decide to spend millions sorting the place right out. But it's impact on the Merlin parks that remain would likely be negligible (after the loss of MAP sales as a result of loosing their biggest name attraction has been taken into account of course). They'd likely increase their attendance by taking custom away from alternative leisure activities, or even be responsible for actually increasing national leisure spend.
It may even actually attract some people to check out more parks if they're enjoyed themselves, as was the case when Blackpool, Towers, and Drayton all opened major investments in the same year. All 3 parks likely increased their patronage by enough people in 1994, using the logic being stated, to push many of the other parks below the waterline. But more people just visited more parks, and rode more rollercoasters that year.
It's not direct competition, and that seems to be what is implied time and time again whilst pondering over why Merlin parks are in such a crappy state. Theme parks are not necessities, and have no right to exist. Some countries have none at all. If the product is good enough, and decent value for money, people will come. The market is as big or as small as economic conditions and product value dictate.
Merlin owning so many parks in a single country is likely only a small factor in ever declining standards. Heide Park can be a bit of a bin as well, but Merlin don't own a single other park in Germany. It looks to me like the forces of Competitive Rivalry are actually quite high on Merlin's UK Theme Parks, hence why they feel the need to advertise heavily and promote them in cereal boxes and bottles of hand soap, and are constantly discounting. But that doesn't come from parks owned by different operators a 100 or so miles down the road, but other activities that people can choose to spend their leisure money on. This could be why they trade so down-market. Investing in them is extremely expensive, and very high risk, so they probably accept that if you want high quality you'll just go and find it abroad anyway. Why bother when you can bolt them on to a load of other attractions in your portfolio and bundle them into an annual pass that you can flog the hell out of? Or if you can sell a cheap gate ticket and then upsell an expensive hotel room, fastracks, and a ten quid hotdog on top?
It still looks to me that the Theme Parks are the anchor attractions to flog passes. Without them the MAP is dead. They ride on the coat tails of past glories, and the historical prestige that their names still manage to carry. A conscious decision has been made, informed by market analysis, that investing heavily in them is not compatible with the strategy of the wider business. So they don't.