I think we need to start off here by clearing up some misconceptions about what it means for something to be “observable, testable and repeatable.” Young earthists love to make a lot of noise about this, but they have some serious misconceptions about what “observable, testable and repeatable” actually means.
For starters, it does not mean that you have to have been there right from the beginning to see things happen, and it does not mean that you can just cry “assumptions” as some kind of magic shibboleth if you weren’t. The “were you there?” argument is a lie—a fallacy that preys on non-scientists’ misconceptions and misunderstandings of how science actually works and what it is and isn’t capable of.
“Observable” simply means that you use observation and measurement to determine and verify basic rules and principles that tell us how things work. Principles such as Newton’s laws of motion, Maxwell’s equations, quantum mechanics, radioactive decay laws and so on. These rules and principles can then be expressed as precise mathematical equations, and those mathematical equations can then be used to give us an indication of what to expect in response to specific actions (such as turning on a light switch for example). Those exact same principles can be used to place tight constraints on what could or could not have happened in the past.
“Testable” means that you use the rules you have derived through observation to make mathematically precise predictions about what you would expect to see if a theory were correct (or wrong) and then make further observations and measurements to check whether that is what you actually see in reality. For example, the Big Bang theory predicts that we should see a uniform microwave background radiation in the sky corresponding to a temperature of 2.7 Kelvin. This was first observed in 1964 by Arno Penzias and Robert Wilson, for which they won the Nobel Prize in 1978.
A young earth, likewise, makes testable predictions: it predicts that we should have sequenced the entire T-Rex genome by now; it predicts that we should never see more than a few parts per billion of lead in zircon crystals; it predicts that we should not see any stars more than six thousand light years away; and so on and so forth. Every single mathematically precise prediction made by young earth models has failed spectacularly while the only “predictions” that it does make that can be claimed to have succeeded are too broad and imprecise to prove anything.
“Repeatable” means that when two or more scientific methods or measurement techniques give the same mathematically precise results, despite being made in different circumstances or at different times, or making different assumptions, that is evidence that those results are, in fact, correct. For example when radiometric dating tells us that the Hawaiian islands have been moving over a hotspot in the Earth’s mantle for the past 80 million years at a rate of 8.6 millimetres per year, and when direct GPS and laser-based measurements give us exactly the same result in the present day, that is repeatability.
So in a nutshell: the claim that the Big Bang is not observable, testable and repeatable is very, very misleading at best. The event itself may not be directly observable or repeatable, but it has left behind evidence, and that evidence must be interpreted according to strict rules and principles that very much are observable, testable and repeatable. This being the case, the claim that it is not scientific is simply not true and demonstrates a serious lack of understanding of what science is, how it works, how it is conducted, and what it is capable of.