Show that, if $t_{3n}=x$, $t_{3n+1} = y$ and $t_{3n+2} = z$ for all values of $n$, then $p^3+q^3+3pq-1=0$ 
A sequence of numbers $t_0,t_1,t_2,...$ satisfies  $$t_{n+2} = pt_{n+1} + qt_n, ~~~n\geq0$$
where $p$ and $q$ are real. Throughout this question, $x,y,z$ are non-zero real numbers.
Show that, if $t_{3n}=x$, $t_{3n+1} = y$ and $t_{3n+2} = z$ for all values of $n$, then $$p^3+q^3+3pq-1=0~~~~~~(*)$$
Deduce that either either $p+q+1=0$ or $(p-q)^2+(p+1)^2+(q+1)^2=0$. Hence show that either $x=y=z$  or $x+y+z=0$

I have attempted this question, but despite arriving at (*), I don't believe I did it in the correct way.
Here are my workings:
When $n+2 = 3p, 3p+1, 3p+2$ we get three equations.
$$\ z=py+qx\\ x=pz+qy \\
 y=px+qz$$
Adding all three equations, we get, $$x+y+z = p(x+y+z) + q(x+y+z) \\ \therefore x+y+z \not=0 \implies p+q=1$$
I then cubed both sides of this equation resulting in $p^3+q^3+3p^2+3pq^2=1 \implies p^3+q^3+3pq(p+q)-1=0 \implies p^3+q^3+3pq-1=0$
then we can proceed to factor this expression to get $p+q=1$ or the sum of the squared terms.
I suspect this approach is incorrect, but not entirely sure why it's incorrect. I would appreciate an explanation on the matter.

For a shortcut, note that the characteristic polynomial of the linear recurrence is $\,x^2-px-q\,$, but the additional conditions also require $\,t_{n+3}=t_n\,$ with characteristic polynomial $\,x^3-1\,$. It follows that they must have a common root, so $\,0 = \text{res}(x^3-1, x^2 - p x - q)=-p^3 - 3 p q - q^3 + 1\,$.
Now back to OP's actual question.

I suspect this approach is incorrect, but not entirely sure why it's incorrect.

It is incorrect because of the assumption $x+y+z \ne 0$ made earlier, which is not given in the problem. This does not render the attempt useless, but leaves it incomplete. Going back to what's being asked:
$$
\begin{align}
p^3+q^3+3pq-1=0
\\ \iff \quad\quad\quad\quad\; (p + q - 1) (p^2 - p q + p + q^2 + q + 1) = 0
\\ \iff \quad\quad (p + q - 1)\big((p-q)^2+(p+1)^2+(q+1)^2\big) = 0\tag{*}
\end{align}
$$
You proved that if $\,x+y+z \ne 0\,$ then $\,p+q-1=0\,$, which is sufficient for $\,(*)\,$ to hold. However, what remains to still be proved is that if $\,x+y+z = 0\,$ then either factor of $\,(*)\,$ is zero.
For that, substitute $\,z=-x-y\,$ $\, \iff t_2 = -t_1 - t_0\,$ in the recurrence relations for $\,n=0, 1\,$:
$$
\begin{cases}
\begin{align}
-t_1 - t_0 = t_2 &= pt_1 +q t_0
\\ \iff \quad\quad (p+1)t_1+(q+1)t_0 &= 0 \tag{1}
\\ t_0 = t_3 &= pt_2 +q t_1 = p (pt_1 +q t_0 ) + q t_1
\\ \iff \quad (p^2+q)t_1+(pq-1)t_0 &= 0 \tag{2}
\end{align}
\end{cases}
$$
Regarding $\,(1),(2)\,$ as a homogeneous linear system in $\,t_0,t_1\,$ which is known to have non-trivial solutions since $\,x,y,z \ne 0\,$, then it follows its determinant must be zero, proving $\,(*)\,$:
$$
0 = 
\begin{vmatrix}
   p+1 & q+1
\\ p^2 + q & pq -1 
\end{vmatrix}
= -\big(p^2 - p q + p + q^2 + q + 1\big)
$$
