I don't see how this rule could be stated: "we allow claims that in the opinion of the OP are believed by many people, but they never stated explicitly in writing"?
No, claims need to be explicit.
There are a few reasons for this but the main motivation is that we want to avoid debating straw men.
More in detail, and expanding on my comment on Nomen Agentis' answer, there are two main possibilities. There is the case in which an implied claim is accompanied to direct versions of it, in which case I hope we all agree that direct claims are better. This does not seem to be the point of your question of course.
We are left with the case in which there are only people "suggesting" or "implying" that a claim is what it is. My objections here are multiple.
First of all, I posit that no one truly believes the claim and in fact it is merely a hypothesis. Looking at your example, there's nothing disprovable, in saying that a monster fish might be the result of nuclear contamination: it's an opinion and the lack of evidence is admitted; adding an answer which confirms this is not very valuable.
Secondly, we have no way of agreeing on whether a particular implication is correct. This is equivalent to allowing an open discussion on a chosen topic, or to allow people to close questions arbitrarily. Imagine people asking a question on an implied claim and others voting to close because they think the implied claim is different and the question is thus invalid. That is not going to work: good site rules need to be objective and not subjective.
There is nothing wrong with debates, but the Stack Exchange platform is notoriously built as a scalpel to attack a specific set of problems. Allowing debate is not something that works well on this platform.
Thirdly, it's unclear what would constitute an implied claim. If Tom says that Dick believes that cheese can cure cancer, but they don't, is this an implied claim that cheese is anticarcinogenic? In my opinion it is quite the opposite. If someone throws a saucer in the air and takes a picture that looks like a UFO, then posts it on their blog saying "A picture of an actual flying saucer", I do not know how do we determine if they are being ironic or fraudulent.
I think that asking a claim to be explicit is a good rule, that works well with notability. In fact, I'd say that notability implies to be able to find an explicit claim, since we ask that a valid claim be a relatively common belief and I don't see how that can be true if no one ever states it explicitly. If one struggles to find an explicit version of a claim, it might be the case that it is not notable -- in fact if the OP themselves do not believe the claim, it is very likely this is the case.