Of course little things like the equality of inertial and gravitational mass (so that objects of different density fall at the same speed) don’t fit easily into such a picture.
Also, the rate at which the hypothetical particles collide with matter has to be extremely large in order to work with very dense matter, such as free neutrons.
Neutrons have been observed to follow Newtonian gravity to very high accuracy in the lab.
And then there’s the complete absence of Brownian motion in free particles. With some huge flux of particles carrying the sort of momentum that would be required to account for the gravitational motion of free neutrons, the resulting fluctuations would be very visible.
Besides, if the particles bounce off the gravitating objects, their velocity distribution will change as a consequence. (Some of them will rattle around between them, going faster and faster as the objects get closer.) Thus there will be a wake effect, like a small plane taking off right after an A380. No such effects are observed.
Not to pile on, or at least not as much as the notion deserves, but if relativity is completely wrong, then there is only one velocity in a given reference frame for which the drag force of such a particle ensemble is zero.
And, of course, there’s the question of the origin, distribution, and regulation of the momentum-carrying particles.
To have any chance of avoiding even these purely classical effects, the particles would have to have infinite speed, zero mass, perfectly uniform and isotropic distribution in both position and direction, perfectly timed arrival at each object to make the fluctuations cancel out, and on an on.
This is the luminiferous ether, on stilts.
And then there are matter-wave interferometers, which work not only on electrons, but on neutrons and even buckyballs. They set far tighter limits on most of these classical effects.
So no, these sorts of theories are not good candidates to explain gravity or other relativistic effects.
Cheers
Phil Hobbs