There's a mechanism that would make photons behave as "tired light" without even interacting, just by the mere presence of matter or energy for the spacetime intervals they have to cover. And the longer the distance, the more redshifted the photon will get.
To explain how this works, we have to think about the double slit experiment.
In a double slit experiment, photons end up creating an interference pattern on the screen because the presence of the wall with the slits modifies how the photons will "explore" the free space between the light source and the final screen.
But we must realize that the photons that end up shaping the interference pattern on the screen never interact directly with the wall, and yet they create a pattern that is different to the pattern they'd have created if the wall were not there.
We need the wall to get the interference pattern, but the pattern emerges, precisely, because the photons that reach the screen didn't interact with the wall.
This means that the mere presence (without direct interaction) of the wall causes the photons to interfere with themselves, changing how they arrive to the screen. By not interacting with the wall, the photons end up tracing a different geodesic than the one they’d have traced if the wall were not there.
Now we can extrapolate this example to what happens to every photon in outer space whenever it faces a particle in the intergalactic medium.
Traversing a distance of a billion lightyears through the intergalactic medium without ever interacting is like facing billions of double slit experiments in a row for a photon. The longer the distance, the more particles the photon has to avoid interacting with, which causes it to self-interfere and change its wavefront a bit.
This way cosmological redshift would be just regular gravitational redshift, but caused by sparse matter distributions over long distances instead of dense matter distributions over short distances.
The average energy density of the universe causes tiny distortions and delays in the photon's wavefronts as they "fly" through vacuum avoiding particles.
These self-interference effects build up with distance, and that's what explains cosmological redshift and the value of the Hubble constant. It'd be related to the average density of the universe, not to a physical expansion of space.
The self-interference effects that matter particles impose on photons by the mere fact of existing constrain the paths photons can take to keep travelling through free space.
This "tired light" mechanism doesn't cause blurring because it comes, precisely, from non-interaction. The photons accumulate delays and distortions, but they are the same photons that leaved the light source. They never interacted with anything else during the trip and never leaved empty space, only that they got a bit redshifted each time they passed by a matter particle and didn't interact with it.
