Why can’t you simulate a Turing machine with a one-stack PDA by messing with the stack?

I have heard that a matrix can be modeled as just an one array by declaring increasingly large spaces to be from the second array, and that the least you need for a Turing machine is just a PDA with two stacks. I also know that you can’t simulate a Turing machine with a PDA, because a professor challenged a student when he suggested the treating a stack as a matrix, that is, as if it were two arrays. What’s the roadblock?

• The stack can only be accessed in a very particular way. Using two stacks you can simulate a Turing machine, however. – Yuval Filmus Nov 7 at 19:50

Now since CFL is a proper subset of CSL which is again a proper subset of recursive languages which can all be computed with Turing machines, Turing machines can simulate all PDAs but not vise versa. To see why, for a given recursive but not context-free language, say the standard example $$a^nb^nc^n$$, there is no PDA to compute this language since else there would be a context free grammar to compute it which would make it CFL. On the other hand it is not hard to build a Turing machine to compute this language.