Why doesn't this calculation work? I want to find some closed form for $\gcd(x^3+1,3x^2 + 3x  + 1)$ but get $7$ which is not always true.

With your procedure you found that the GCD between the two polynomials $x^3+1$ and $3x^2+3x+1$ in $\mathbb{Q}[x]$ is $7$, or equivalently $1$, because the GCD of polynomials is defined up to constants (every scalar value $c$ divides any polynomial $p(x)\in\mathbb{Q}[x]$). 
Thus there is not contradiction in your statement.
