What's wrong with manipulating this algebraic equation? and why does a manipulated system of equations have a different solution than the original?

The problem is that $x+1$ is not in general equal to $-\frac1x$: that equality holds specifically for solutions to the original equation $x^2+x+1=0$. Thus, the new equation is not in general equivalent to the original quadratic: they are equal only when $x$ is already a solution to the original quadratic. Since $x=1$ is not such a solution, the fact that it is a solution to the new equation is irrelevant to the original problem.

In effect you multiplied by $x-1$ when you converted the original quadratic to a cubic, thereby introducing the extraneous solution $x=1$: $$(x-1)(x^2+x+1)=x^3-1\,,$$ so $(x-1)(x^2+x+1)=0$ iff $x^3-1=0$.

You may not substitute parts of an equation into itself. E.g. if $x^2=x$, the substitution of $x^2$ by $x$, gives $x=x$ !