Also cross-posted on math.SE

The basic CRC algorithm is very straightforward (cf. this answer of mine on math.SE) but actual implementations have various bells and whistles included. Thus, unless you have the details of these bells and whistles, your reverse-engineering will fail. What you do have in most cases is the actual payload of the message sent available to you in the clear (no encryption) but what the motor does with the commands it has been sent, or what it sends back as an ACK or a response to the message, is something that we have no information about.

Your data are inconsistent. A quadratic polynomial with zeroes at $2$ and $5$ must satisfy symmetry requirements $p(3)=p(4)$ and $p(1)=p(6)$ which what you have written down fails to do.

Since $23$ is the secret, it should be smaller that the field size, no?

You are overloading the meaning of p and $p$. p is a prime number while $p$ denotes a polynomial. Please edit your question to clarify what you mean.

It is good that the OP does not know how to prove the general statement, because the general statement is false.

For $k>1$, the integers modulo $p^k$ are not isomorphic to GF$(p^k)$; for $k=1$, they are. That's all there is to it.

@Mathys Thank you for your careful reading of my answer and correction of a typo that nobody else noticed for 11 years!

If you must use the ring $\mathbb Z_{2^n}$ instead of the field $\mathbb F_{2^n}$ as secret-sharing schemes usually do, then all bets are off: secret-sharing schemes over rings have many flaws that make them unusable in practice. If you are wiling to use a field (not necessarily $\mathbb F_{2^n}$; any large enough finite field will do), then @CommandMaster's comment is right on target. See this answer re the necessity of using a field and this one for how RS coding can identify cheaters.