That's a bit of an oversimplification that leaves out important qualifiers. "Non-significant" usually means it's indistinguishable from a null hypothesis when a certain level of randomness is allowed in a single, isolated trial.
Who picks the null hypothesis? How is it picked, i.e. why does that specific hypothesis get favourable treatment? What level of randomness should one allow? What does it even mean for an experiment to be a single, isolated trial? How can anything be?
Those are critical questions to understand the concept, and your explanation just pretends they don't exist.
That's another fallacy of frequentist reasoning, that we have to draw definitive conclusions from evidence. That something is definitely false until we have "statistical significance" where it all of a sudden becomes definitely true.
In real life, to borrow your description, we can hold varying levels of belief in statements depending on how strong the evidence is, and the magnitude of the payoff in the various cases.
Maybe the probability of the result in the study in question is 51 %. That's still more than 50 %. Whether that difference is meaningful to you is not something someone else can decide.
Nobody who knows what they are doing, and uses statistics, can flip from something being definitely true to definitely false. At best, they can find overwhelmingly convincing probabilities close to 0 or 1.
Honest scientists who use statistics do not make such a claim that an effect does not exist. Rather than the experiment that was conducted did not produce sufficient evidence (to a numerically defined standard) which justifies believing in the effect.
That is to say, that the existence of the effect, given the results of the experiment, has a low likelihood, and that low likelihood can be statistically quantified.
What that means is that exactly the same results as were observed will, or would, with a high probability, also be observed if the experiment occurs in the null hypothesis universe: the world in which the effect is absent.
So even if we are not in that universe (the effect is real), the experiment didn't show it.
The experiment simply doesn't discriminate between the null hypothesis and its negation to a level that could convince one to hold a probabilistic belief in the existence of the effect.
> the existence of the effect, given the results of the experiment, has a low likelihood, and that low likelihood can be statistically quantified
You have this completely backwards. It means that the likelihood of the null hypothesis was not below some threshold such that it can be "ruled out". It says absolutely nothing about the likelihood of the data if the effect exists.
Of course, but the fact that people apply a binary threshold tells you that they want to be able to rule out some things from their models entirely, and include other things as something that's as good as a true fact.
What does a non-binary threshold look like, and how is it different from just fine-tuning a regular binary threshold to err more or less on the side of caution?
Who picks the null hypothesis? How is it picked, i.e. why does that specific hypothesis get favourable treatment? What level of randomness should one allow? What does it even mean for an experiment to be a single, isolated trial? How can anything be?
Those are critical questions to understand the concept, and your explanation just pretends they don't exist.