Need for random bits in final PCP theorem statement
The statement of the PCP theorem is essentially equivalent to the following statement:


  For every problem in NP there is a reduction $f$ to $O(1)$-FUNCTION-SAT such that if $x$ is a YES instance then $f(x)$ is satisfiable, whereas if $x$ is a NO instance then at most half of the clauses of $f(x)$ can be satisfied.


Here an instance of $k$-FUNCTION-SAT consists of a collection of "clauses", each an arbitrary function on $k$ inputs.

Where did the parameters $O(\log n),O(1)$ in the PCP theorem disappear? The first parameter is the logarithm of the number of clauses, so there are $2^{O(\log n)}$ clauses; the second parameter is the arity of the clauses. Together, these parameters guarantee that the FUNCTION-SAT instance has polynomial size, and so the reduction runs in polynomial time.

PCPs are not always described this way, but the descriptions are equivalent. Proof of the equivalence left to the reader.

You are enumerating over all choices of $O(\log n)$ when you are constructing the FUNCTION-SAT instance. But it only allows you to construct the instance, not to tell whether it is satisfiable or not; that is an NP task.