You're analysis is wrong right from the start.

You said:

Quote - Instead of saying there is a car and two goats, let's get abstract and say there is an A, a B, and a C.  You always pick door 1 and C is always revealed. 

You left out the case where door 1 is C. In that case it is not revealed, because Monty never shows you what's behind the door you picked.. He reveals B instead. You said that that case was excluded from the problem, but it can't be excluded because that is the third case. This is the crux of the decision that Monty makes, and that you are not aware of.

The ABC thing is really leading you down the wrong path.

The way that most people finally understand this is to extend the problem. Restated, there are N doors. Exactly one has the prize. The rest are losers. Try listing all the scenarios for 1000 doors. That's beyond what anybody could do in a lifetime. You have to look at it differently.

So in the general situation where there are N doors...

You pick a door. The chance that you have the right door is 1 out of N.

Monty carefully opens N-2 doors, leaving only two closed doors. While doing this he makes sure that since he knows where the prize is, he does not choose that one. This is the key to the problem.

He then asks you to stay or switch.

With this restatement, it should be obvious what is going on.

When Monty picked all those doors to open, he was avoiding the prize. One out of N times, your choice was the right door and it didn't matter what he opened. But (N-1)/N times the door he didn't open is the prize.

Let's say there are 1000 doors.

You pick door #1. You have only 1 chance in 1000 that you're right. Most of the time, you're wrong. He opens 998 doors no matter what, all the while carefully avoiding the prize. There are two doors left - yours and the one he decided not to open, which more than likely is the prize. In fact, the chance that the remaining door that neither you nor Monty chose is the prize is 999 out of 1000.

So back to the original with 3 doors.

You pick door #1. You have only 1 chance in 3 that you're right. Monty opens one door, carefully avoiding the prize in the situation where you're wrong. That is 2/3 of the time. So 2/3 of the time the remaining door that neither you nor Monty chose is the prize.

The key is to understand that you know nothing and are choosing randomly. Monty knows everything and is not choosing randomly.

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Renderosity forum reply notifications are wonky. If I read a follow-up in a thread, but I don't myself reply, then notifications no longer happen AT ALL on that thread. So if I seem to be ignoring a question, that's why. (Updated September 23, 2019)