> alpha = (4GM)/c^2*r
r has units of length.
A geometrical space in which closeness is defined and measurable by a numerical distance -- in this case r -- is a Riemannian manifold. "c" is a dimensional physical constant with units LT^-1 or length per time <https://en.wikipedia.org/wiki/Dimensional_analysis#Simple_ca...>. "Time" enters into the manifold making it semi-Riemannian or pseudo-Riemannian, and however you solve your equation, it can be represented in a pseudo-Riemannian manifold. If there are 3 spatial and 1 time dimension, the manifold is Lorentzian <https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold#Lor...>. Lorentzian manifolds are the spacetimes of relativistic physical theories.
Newton's big G incorporates length and duration; "4M" is the only term in your equation which includes neither.
So it seems less like you have a problem with spacetime (having shown you're OK with length and duration), than that gravitation is identified with spacetime curvature, and that this is captured by the metric tensor which encodes how lengths, durations, and angles change with tiny displacements from a point on the manifold.
> physicists decided that spacetime ... is emergent
Nobody's decided that, but some physicists are chasing that as a hypothesis. The idea is that there is some underlying theory that recovers equations like the one you quote in some low-energy limit of a theory that is not set in a smooth pseudo-Riemannian manifold.
See, "every smooth manifold admits a Riemannian metric" <https://en.wikipedia.org/wiki/Riemannian_manifold#Every_smoo...>. Smoothness is really a question of "no sharp corners", and tends to be expressed in terms like "everywhere differentiable". One can imagine a spacetime that is made up of spacetime-tetrahedra (you and three friends run together towards a point) and see that the triangular features can be non-smooth when you look close but become smooth as you put lots and lots of tetrahedra together and look at them in bulk. (Loop quantum gravity considers something like this).
However, practically all the differential equations we use in practice rely upon smoothness, and modern ones (since the 20th century introduced tensors and general covariance) would break if the universe around us were not very very very very similar to a Lorentzian manifold.
There are alternatives to curvature of the Lorentzian manifold (i.e., they keep special relativity and its flat spacetime) for encoding the gravitational interaction, with various trade-offs and exposure to narrowing limits obtained from astronomical observation. Gauge theory gravity (GTG) is an example of one that is in good shape.
> people who said these things in forums would be dubbed crackpots
I'd reserve crackpot for someone who pesters scientific journals and working scientists (yeesh, some will call you at home) with their non-standard and often quite verbose ideas even after they are shown conclusively to be wrong in some way. Especially if they think the non-standard idea will revolutionise the field. Pursuing a mathematically-expressable theory from which General Relativity (or something very close to it) emerges in some limit (and Newton/Poisson/Gauss gravity in the low-speed low-escape-velocity local limit of that) isn't cranky. Insisting that the new theory is world-changingly revolutionary probably is cranky, but at least they will have shown they understand existing theory and how theirs differs in detail.
What word should be used for someone who denounces an area of research while obviously not understanding the research denounced?
Ignorant? Ignorance is curable. Every working scientist was once an ignorant kid. They read a lot of textook material, got plenty of help from people with more experience, became used to being shown how they misunderstood something or other, and learned how they could rigorously demonstrate how someone else -- including authors and editors of textbooks -- got something wrong. The result: a firehose of papers on arxiv and elsewhere proposing and exploring in detail something that is not the standard model of gravitation (or particle physics, or fluid mechanics, or whatever).