So, then, what brought that UNcaused thing into existence?
Your premise creates the problem
I think the problem existed way before I did.
The problem occurs IF you state that "nothing can exist that something else brought into existence."
No, that's not what I am saying at all. I am saying that the reason we exist is because we were brought into existence. If the UNcaused "exists" then it, too, must have been brought into existence or else it cannot be said to exist.
This still has the premise that "everything that exists needs a cause." If you take this premise, all you have are dependent causes - the infinite regress, and the result that nothing exists because the "cause" buck is passed on infinitely. Turtles all the way down.
The first cause argument does NOT use this premise, therefore it's not violating itself by having something exist that is uncaused. In the First Cause argument everything in motion needs a cause, everything dependent, changing, in time, imperfect needs a cause. This set includes everything we sense in the universe, everything in space time.
The First Cause is not in motion, independent, outside time, perfect and unchanging. Therefore it is not in the set of that which needs a cause.
Looking at it strictly logically:
Assuming everything must have a cause as a premise, results in nothing existing (all dependent causes lacking the necessary independent cause). The first cause argument avoids this obvious failure.
Second, if you are emphasizing that everything for everything that exists there must be a time when it didn't exist, then we're back at the "eternal = outside time" point where "time" and "when" and "before" etc. are meaningless.
All of this fits together. In order to explain why anything exists, X must be true; if X is not true, nothing exists.
Acquinas developed four variations of his basic argument and they've lasted as a standard to test against. They're pretty tight.