Cook-Levin theorem proof's requirement of $\phi_{cell}$
We are building a reduction from problem Q (some problem in NP) to SAT.  To do that, we construct a function $f$ that maps an instance of Q to an instance of SAT.  In other words, if $w$ is an instance of Q, then $f(w)$ is a formula in CNF form.  (Recall that SAT is a decision problem whose instances are CNF formulas.)  Let's call that formula $\varphi$.  We'll remember that $\varphi$ depends on $w$.

What is the formula $\varphi$?  A formula is something that takes any assignment $x$ and assigns a truth value to it, namely $\varphi(x)$.  The way the reduction works, we interpret $x$ as a tableau, and $\varphi$ is defined so that $\varphi(x)$ will be true if $x$ is a valid tableau for the input $w$ (a valid computation of a non-deterministic Turing machine for problem $Q$ when run on input $w$).

So, $f(w)$ is not the tableau itself.  $f(w)$ is a formula that accepts valid tableaus.

Then, once we have defined $f$ (i.e., $\varphi$), we prove the following property: $\varphi$ will have a satisfying assignment if and only if $w$ is a yes-instance for problem Q.

For this to work out, the formula needs to include terms that ensure that any satisfying assignment to $\varphi$ is a valid tableau.  If you left that part out, you wouldn't be able to prove the property mentioned above, so $f$ wouldn't be a correct reduction.