I already know the well know and correct solution. However want to explore with my intuition and find the flaw in that.
Please help me proceed further with my approach (if at all it leads) to solution or let me know the flaw.
Hypothesis : The car is behind un-chosen door
$U1$ - User chooses gate with car
$U2$ - User chooses gate with goat
$P(U1) = 1/3$
$P(U2) = 2/3$
$P(U1|H) = 0$
$P(H) = 1/3$
$P(U2|H) = 1$ --> Probability of $U2$ when H is known to happen is 1; since gate with car was not selected by user earlier, the goat is only possible choice.
$P(H|U1) = 0$
$P(H|U2) = ((1/3).1) / (2/3) = 1/2$
Is it possible to take this further to correct solution.
$\mathsf P(H)=2/3$, not $1/3$
Substituting $C\gets U1$ and $G\gets U2$ to be a little more self-commenting.
$${\begin{array}{|l|l|}\hline \mathsf P(C)=1/3 & \mathsf P(H\mid C)=0 & \mathsf P(C\mid H)=0 & \mathsf P(C\cap H)=0\\[1ex] \hline \mathsf P(G) = 2/3 & \mathsf P(H\mid G)=1 & \mathsf P(G\mid H)=1 & \mathsf P(G\cap H)=2/3 \\[1ex] \hline\end{array}\\[2ex] \mathsf P(H) ~{= \mathsf P(G\cap H)+\mathsf P(C\cap H) \\[1ex] = 2/3} }$$
When the user chooses the gate hiding the car, then switching will certainly be a loss. When the user chooses a gate hiding a goat, then switching will certainly be a win. The user has a probability of two thirds for choosing a gate hiding a goat. Therefore the probability that switching will win is $2/3$.
Remember: Monty "cheats" and always reveals one of the goats.