Are fixed points of RG evolution really scale-invariant?

No, dimensionful couplings do not have to be all set to zero at an RG fixed point. An RG fixed point is one where all of the beta functions vanish, and beta functions generally have the form $$\beta(g_i) = (d_i - d) g_i + \hbar A_{ij} g_j + \ldots$$ where $d_i$ is the dimension of the corresponding operator. If one truncates the series at $O(\hbar^0)$ then the only possible solution is to have $g_i = 0$ if $d_i \neq 0$, so in classical field theory the only fixed points are the massless free theory and massless $\phi^4$ theory.

In a quantum field theory we must account for loop diagrams, which give terms that are higher order in $\hbar$. Then the zeroes of the beta functions are completely different; the massless free theory remains a fixed point, called the Gaussian fixed point, but massless $\phi^4$ theory acquires a mass scale by dimensional transmutation. But this process can also work in reverse. In this case there's a new fixed point, the Wilson-Fisher fixed point, where the classical mass term is nonzero. This is dimensional transmutation running in reverse; the mass renormalizes to exactly zero.

If all the couplings are zero then you are sitting on the trivial Gaussian fixed point. Being a fixed point is characterized by the vanishing of the beta functions (some derivative of the couplings as functions of scale), not that of the couplings themselves.

Also, in general, scale invariance does not imply conformal invariance. You need something extra like a traceless energy-momentum tensor. The study of this issue is an active area of research, see the review "Scale invariance vs conformal invariance" by Nakayama.