I am finding a hard time to grasp what you are telling here. If you deduce a set of laws A from an event X, you don't need X to repeat to check the validity of A. You can check it against another event Y, where you predict its characteristics using A. If the laws are valid, the prediction ll match the actual observation..
That is exactly what I wanted to ask you originally when you cited the celestial mechanics example. It is not relevant in cases we are deriving a theory underlying the behavior, rather than axiomizing the very specific behavior itself...
Deriving an underlying theory would be deducing the force of gravity from it.
Axiomizing the very specific behavior would be making a rule that says. "If the object A (and object A only), while moving in a straight line, comes at so and so coordinates with respective to B, will end up in an orbit around it"
So these are two completely orthogonal issues. The question of what you do with the result of an experiment is completely independent of whether or not you can reproduce that result. The latter is what is under discussion here.
So you can, for example, observe that there was a solar eclipse at a particular time and a particular place. But you cannot repeat that observation. (By way of contrast, you can drop and object from a particular height and observe that it takes a particular time to fall. You can repeat that observation.)
What you do with those observations has (almost) nothing to do with whether or not you can repeat them.