Equisatisfiability in the reduction from 4-SAT to 3-SAT
You're right in saying that they should be equisatisfiable. And they are.

I'm not sure why you think converting your unsatisfiable $4-\text{SAT}$ instance into a $3-\text{SAT}$ instance would make it satisfiable. Because it doesn't.

If a $4-\text{SAT}$ instance is unsatisfiable, then no matter how you choose to assign truth values to your variables, there will be some clause that is not satisfied. Without loss of generality, let that clause be $(a \vee b\vee c\vee d)$. Since this clause is not satisfied by your assignment, it follows that all of $a, b, c$ and $d$ are false.

Now converting this into a $3-\text{SAT}$ instance gives you these pair of clauses: $(a \vee b \vee z)(\neg z \vee c\vee d)$. Note that since $a, b, c$ and $d$ are all false, in order for both of these clauses to be satisfied, both $z$ and $\neg z$ need to be true, which is impossible.

In other words, the resulting $3-\text{SAT}$ instance is also unsatisfiable.