checking configuration history of Turing machine using PDA

I am trying to understand the technique of using configuration history in proofs.

To prove that: $ \{<M>|M\,\,\,is\,\,\,a\,\,\,TM\,\,\,and\,\,\,L(M)=\sum^* \}\notin RE$

given $ <M,w>$ we have built a Turing machine that accepts all words except M accepting configuration on w. (and then simple reduction)

To prove that: $ \{<P>|P is\,\,\, a\,\,\, PDA\,\,\, and\,\,\, L(P)=\sum^*\}\notin RE$

we showed the same proof, only that we built a PDA that accepts all the words except the accepting configuration of M on w.

Does PDA’s ability to determine whether input is an accepting configuration of M on w actually means that I can simulate M’s run on w with a PDA? Or testing whether configuration is an accepting configuration is different from a simulation