Can God be proved with logic?

There is a fairly well know argument for the existence of God called the Ontological Argument. It has a lot of forms but generally it goes like this.

1-God is the greatest imaginable being.
2-All else being equal, a being or entity that exists is greater than one that doesn't.
3-Therefore, God exists.

When I first heard this one I thought something like. "This actually convinces people?" It just seemed so ridiculous that you could define God into existence. But yet, It is hard to see what the problem is with the argument, it seems to be logically sound.

Respected theologians like William Lane Craig use this argument all the time, they see it is a slam dunk proof for god.

But it isn't. There are a lot of problems with the argument, I won't go into detail on all of them here. If you are interested there is an entire explanation and breakdown of the argument here. But one easy way to see how this argument is wrong is to try to apply it to other things. The technique is called Reductio ad absurdum. All you need to do is use the exact same logical steps to see if you can get to an absurd conclusion. If you can, then the argument has problems. Although it doesn't necessarily tell you where the problems are. So let's see if we can do that in this case.

1-Shangri-La is the greatest place on earth.
2-A place that exists is greater than one that doesn't.
3-Therefore, Shangri-La exists.

1-Sasquatch is the most fearsome creature imaginable
2-A creature that exists is more fearsome than one that doesn't
3-Therefore Sasquatch exists.

But here is my favorite one

1-God is the greatest thing that can be conceived.
2-Two Gods are greater than God.
3-1 and 2 are in contradiction.

Do you see where the logic breaks down? The trick is that the property of existence is being implied, although not actually stated, in the first premise. So in effect, the Ontological Argument is simply a fancy way of begging the question. Begging the question in logical terms does not mean "brings up the question, like most people use it in normal conversation. It means to assume the conclusion in the premise. Here is a simple example

1-God has done good things for me in my life.
2-Therefore God exists.

The existence of God is being implied in the first premise, otherwise how could he do good things for you?

So how is it that a high-school educated person like me can figure out in less than an hour how ridiculous this argument is, but supposedly intelligent theologians like Craig seem to be convinced by it?