If the Big Bang wasn't brought into existence by free will, then there must be an infinite regress of preceding causes that led to up to the Big Bang.
The Big Bang was either caused or uncaused. The human concept of free-will has no bearing on the matter.
Given an infinite amount of time, any action that has a greater than 0 chance of occurring will inevitably occur.
We have no indication if time itself (or anything else for that matter) is "infinite" or not.
One this action inevitably occurs, we can count backwards the number of trials that led up to the action. By doing this, the entire event has a quantifiable beginning.
The problem is that given an unlimited amount of time,
An unfalsifiable hypothesis, given current human (and possibly fundamentally insurmountable) limitations.
...something that can happen inevitably will. This gives you an infinite chain of events that all have quantifiable beginnings to them which is logically absurd.
Not "logically absurd", in-fact, perfectly logically coherent, except for the hypothetical "first-cause" that may or may not precede the apparent "first cause" which we can currently extrapolate scientifically (which is necessarily beyond our epistemological limits).
The only way around this is if an action occurred by free will...
I'm pretty sure "free-will" has no bearing on this hypothesis. You might be conflating the term "free-will" with "uncaused".
- in that case the event will not have a quantifiable beginning.
If the earliest event we can identify has no known cause, it is essentially and practically, an uncaused event and is yet, very much "quantifiable".
If you are searching for logical evidence of (ontological) god(s), you've found it.
"NTURTTGgTS" = "Noumenon, The Ultimate Reality, The Truth, [G]god, The Source" is a logical and necessary prerequisite of phenomena.
However, if "The Big Bang" is your god(s), you are merely confirming/asserting (ontological) Deism.
And Deism is functionally identical to atheism.