As is well known, the usual Schrödinger equation, $$\mathrm{i}\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^2}{2m}\Delta\psi+V\psi,$$ is not relativistic. It can be derived formally by applying the standard quantization rules $E\to\mathrm{i}\hbar\partial_t$ and $p\to-\mathrm{i}\hbar\vec{\nabla}$ to the non-relativistic total energy relation $E=\frac{p^2}{2m}+V$. However, if we instead start from the relativistic total energy relation $E=\sqrt{m_0c^2+p^2c^2}+V$, we arrive at the relativistic Schrödinger equation $$\mathrm{i}\hbar\frac{\partial}{\partial t}\psi=\sqrt{m_0c^2-\hbar^2c^2\Delta}\psi+V\psi,$$ also known as the spinless Salpeter equation, the (square) root Klein-Gordon equation, and so on. While the appearance of an operator inside a square root may seem nonsensical at first, there are valid methods to give it meaning, coming from functional analysis in the form of pseudo-differential operators.
To the best of my knowledge, it can be shown that this equation does not disturb the structure of the light cone, making it relativistic (in a sense). But I really cannot wrap my head around how it can actually be relativistic? Not only is it non-local, it doesn't even treat time and space the same (which is why I've also encountered it under the name pseudo-relativistic Schrödinger equation).
On the other hand, there is a considerable body of work dealing with this equation and producing (apparently) useful results. So what's the benefit of this equation? Is it a viable relativistic quantum evolution equation? And would it be possible, for example, to construct a "QFT replacement" based on this kind of equation?
Edit: To rephrase my question, is there a compelling physical reason to not use this equation as a starting point for relativistic quantum mechanics? Is it for historical reasons and the square root of an operator, or is it just not feasible? Most questions deal with mathematical or physical aspects of the equation (which I'm aware of), but I'd like to understand why it's not used more.
