Interesting question. The first thing that comes to mind and it’s close to what you describe, is a Bayesian kind of approach.
You can imagine having a non-parametric data distribution as your likelihood and then your hypothetical value can become your prior (pseudo or not) distribution. Combining those two, you will end up with a posterior distribution which you can then compare to your hypothetical prior that you had in the first place, and test if these two are the same or test if the prior is contained inside the “main body” of the posterior.
Example: let’s say that you believe that the average height is 1.78m. You can create a hypothetical distribution as your prior for which “most if its mass” is concentrated around 1.78m. Afterwards, you can combine it with the data distribution from your sample and test if the resulting distribution is not significantly different than your prior. The way I would choose to test it is via simulation and graphs! E.g: simulate 1000 values from the prior and overlay them on top of a boxplot from simulated values coming from the posterior.
Not sure if I’m describing something that people are doing already but to me sounds interesting.