There can be non-experimental evidence that can be taken into account. If the textual evidence goes through peer-reviewed and is deemed credible, then scholars can say that there was a man named Ceasar although we cannot be 100% confident that there was one; there is going to be uncertainty since it relies on the credibility of historians.
Then you concede two key points: that forms of evidence other than experimental evidence can be used to make reasonable conclusions, and that the null hypothesis is not determinatively applicable to situations that do not involve experimental data.
You're still relying on the fallacious idea of there being a first cause.
I think what you really mean here is that I'm relying on the idea that there
must be a first cause.
While studies in quantum mechanics indicates that causality is not a universal truth, in the context of cosmological origins it is necessarily true that there was a first cause, and illogical to conclude otherwise.
Presumably, you hold that the universe was generated through the culmination of random and uncaused activity by some type of substance. Because only the simplest of particles have been observed demonstrating this type of random activity, we have only them to refer to as the appropriate type of substance involved in this phenomenon. Thus, for the logic of this explanation to hold up, we must hold that the existence of these simple particles, and these particles alone, preceded any other type of cosmological development.
For these particles to be capable of preforming the process necessary to initiate new developments they must be capable of motion, that being the act of transition, whether it be subtle or otherwise. On this basis we can draw two possible inferences regarding the state of these particles prior to the development of the universe. Either none of the particle displayed motion, or at least some of them did. However, the idea of either of these possibilities could occur without a first cause is problematic because of the following...
If none of the particles displayed motion, a state static motionlessness, they would be in complete defiance of scientific principles. All matter by definition is characterized by motion. If "matter" is motionless (characterized by a reading of point zero on the kelvin scale) then it is actually not matter at all because it has no energy, and cannot be said to exist. At that point, the "matter" simply becomes a void. I should note that despite the efforts of researchers, no substance has ever been reduced to a state of no motion/point zero kelvin. Thus, if all of the particles lacked motion then they would not be particles at all, or more practically, they could not possibly initiate the universe into being because they wouldn't exist in the first place. Something would have to first bring them into existence, but this would represent a first cause. In it's entirety, the concept that none of the particles displayed any motion, and on there own initiated the universe is implausible.
Now, if at least some of the particles displayed motion, and it seems that they would have to, then we can introduce the concept of time. Under its most mundane definition, time is a measurable interval between two transient events. If displacement occurs, which it does when there is motion, you have two transient events and you can measure the interval of duration between when each event took place. Therefore, if you have motion you also have to have time.
Now this is where it gets tricky. If you have time without an initial cause then you must conclude that there has been an infinite number of events preceding the present. However, this introduces a paradox. If there are an infinite number of events leading up to the present, then that means you must cross over an infinite number of events in order to reach the present. This is of course impossible. The only way to reconcile this is to reject the concept of infinity. However, if you do that, then you create a constraint that necessarily require a first cause, and a last event.
Thus, at least under our current understanding of the matter, the idea that the universe could exist without a first cause is invalid, in spite of the fact that some events can occur without cause.