I often hear that while QFT is a relativistic theory, standard quantum mechanics is not. But fields aren't inherently relativistic, you can easily construct non-relativistic QFTs, a relativistic QFT is one with a Lorentz invariant Lagrangian.
In the same way, it doesn't seem to me that there is anything inherently non-relativistic about the standard description of quantum mechanics, and that the actual non-relativistic part is the Schrödinger equation. As a matter of fact if we substitute the Schrödinger equation axiom with dynamics described purely by quantum channels, with the only requirement that an evolved quantum state is still a quantum state and that the theory is linear, we do recover some statements that resemble locality in the Lorentz sense, like the no communication theorem.
Are fields necessary to obtain a relativistic theory? To what extent is standard QM non-relativistic? If QM is inherently non-relativistic even without the Galilean dynamics given by the Schrödinger equation, why should one care about non-locality in Bell inequalities and "spooky action at a distance"? Those would be expected in a non-relativistic theory, just like the gravitation of the Sun on the Earth is spooky action at a distance in Newtonian gravity.