Proof that POSITIVE-3-SAT is in the complexity class P
It looks like you misunderstood a bit what you should prove since you missed the much easier way to prove although your (trivial) proof looks correct.

An easier proof

Let us call a boolean formula in conjunctive normal form in which all clauses consist of exactly 3 literals and in which no variable appears negative a positive&#8209;3 formula. So

$$\text{POSITIVE-3-SAT} = \{\langle\phi\rangle \mid \phi\text{ is a satisfiable positive-3 formula}\}.$$

Note that any positive&#8209;3 formula is satisfiable since it becomes TRUE if a truth value of TRUE is assigned to all variables.  So

$$\text{POSITIVE-3-SAT} = \{\langle\phi\rangle \mid \phi\text{ is a positive-3 formula}\}.$$

Here is an algorithm $\mathcal A$. Given an input string $s$, $\mathcal A$ just verifies whether $s$ is the encoded form of a positive-3 formula. If $s$ is, $\mathcal A$ returns yes; otherwise, $\mathcal A$ returns no. That verification step can be done in linear time in terms of the number of clauses in $s$. So $\text{POSITIVE-3-SAT}$ is in P.



The moral of the above proof is that you can use whatever known to you to construct the algorithm. All you need is to specify an algorithm and show that it works as wanted. 

It depends on the requirement/convention/stage of your course or the intention of your course instructor whether you needs to explain in detail why that verification can be done in linear time or polynomial time. It looks like that verification step is totally ignored by the proof in the question, which is, in fact, a general accepted practice in the field.