“According to your argument, then god had to have a cause. If everything that exists has a cause or a beginning then if a God exists then it had to have a beginning”
Incorrect. Everything that exists under the laws of our universe has a cause. The law does not apply to things that transcend the universe, such as numbers and other mathematical constructs. These things we understand to be eternal, because they transcend the universe.
God is understood to transcend the universe.
Speculation implies that we pluck an answer out of thin air. On the contrary, we can analyze what we do know, similar to how historians piece together fragments to build an accurate picture of the past. I think the fine-tuning alone allows us to make the logical assumption of intelligent design. If you came across a fully functioning plane in a forest, you would not conclude that such a masterfully engineered object could have just come together by chance. You would apprehend that planning went into building it. Looking at how carefully balanced our existence in this universe is, I find God to be far more likely than no God.
Also, if you postulate that everything has a cause, then the chain of causality must extend infinitely into the past. But an infinite series cannot exist in actuality, so a first cause, or Uncaused Cause, must exist.