We cab choose to believe or not but only 'God' can prove his own existence to humans. So the amount of debate about whether he exists, proves that either:
You are quite right, only God can prove that God exists.
He does not have the ability to prove his existence.
He does not exist outside of believers minds.
He has decided that humans are not worth proving his existence to.
He prefers to keep us guessing because he's shy.
There are other possibilities but in his continued absence we can only surmise.
IF and this is a big IF, God sees that faith is the KEY attribute that he looks for then proving his existence in a way that doesn't require faith may be outside his "master plan".
It could be that God proving his existence beyond doubt is NOT the point of God actually exisiting.
Revelation warns us that even after 1000 years of knowing God and Jesus, when Satan is let lose again, many will still follow Him against God.
There may be indeed a very good reason why an allknowing being chooses to not make its precense known in a 100% proved and factual way so as to convince ALL that he exists.
Perhaps that upon his "revelation" beyond any doubt, things will change forever and that may not be in the best interest of human evolution.
Perhaps the pinacle of human evolution is "to be with God" and that comes not view God revealing himself ( doing the work for us) bit by us "finding him".