Whether what you say about Monte-Carlo depends upon what
you are trying to do. In this case, absolutely. If you are using a Monte Carlo algorithm to calculate a probability, you are certainly better off trying to use
statistically uniformly distributed numbers than really
random numbers for exactly the reason you say. (Speed of
convergence.) Of course for the same problem you are even better off trying to turn it into an integration problem and then attempting standard numerical integration techniques. (Of course we have better means of calculating Pi, but I digress.)
However if you are going to do a large number of complex scenarios which involve multiple random decisions, and particularly if you will then compute summary statistics on those runs, then speed of convergence or no, it is probably safer to use random data for your random decisions.
On a related note, I remember having seen some research showing that chaotic systems can be surprisingly good at detecting pseudorandom input. So again if you are doing a Monte Carlo simulation of how a chaotic system will react, you are not guaranteed of accurate results from using pseudorandom numbers.
So to summarize, for simple problems you are right that the right pseudorandom sequence tends to converge more rapidly. But using good random data can prevent a variety of causes of spurious results.