No, it assumes that relativity of ANY kind is basically correct (Einsteinian, Lorentzian, Maxwellian, take your pick) and that the normal laws of motion are at least slightly applicable to starships. Special relativity doesn't apply to warp drive because warp fields represent a non-inertial reference frame and would therefore fall into General Relativity.
Where did you come up with the assumption or idea that warp travel is in any way non-inertial? How would you explain away the many canon references to the contrary? In the TOS episode "Arena" the Enterprise bridge crew is practically "thrown over the rails" when the Metrons force them to decelerate, hard, from Warp Factor Eight. And again, the inertial effects shown while Kirk and co. execute warp speed evasive maneuvers when Trelane chases after them with the planet Gothos. And more specifically the inertia damper failure cited by Decker while the Enteprise is dodging V'Ger's plasma bolts at Warp Factor Seven.
Measured relative to WHAT? The center of the galaxy? The planet they left? The luminiferous aether?
There's this thing called
drag.. when you get to "Starship Bumpers" later on in my response, I think you will have a better idea about what I am referring to.
That's kind of how it's depicted because the writers Did Not Do The Research, but it makes a lot more sense if you simply retcon it as a measure of acceleration and the exact flight dynamics of warp travel are simply not fully or accurately described (which, either way, they're NOT).
You make a lot of assumptions that writers failed to do their research, or didn't care one way or the other. Perhaps they were under the gun, writing a show for a network that expected results and ratings, with other deadlines looming, all the while trying to entertain the likes of you and me by telling a story. BTW I hate unnecessary retcons that IMHO upend the whole of any narrative world simply because it would make more sense to any one person, be it you, a director, producer, or another writer. To me, it's more than a little arrogant to assume you know all the reasons why something was written in a particular way, and then condemn the work/folks who didn't do it the way you would have.
As it stands, a single particle of gas moving at the speed of light (hydrogen or helium atoms, let's say) is not actually that big of an issue. Conventional space ships with no shielding get hit by those all the time in the form of alpha and beta radiation. Cosmic radiation is believed to be lighter particles at much higher energies. Traveling at very high FTL velocities would simply add energies to those particles and they would strike the hull as a constant rain of high-energy radiation, and a shielded starship is fully capable of just shrugging that off without a problem.
I'm going to give you some reading material to consider. First off is
Rick Robinson's First Law of Space Combat
which gives some idea of the kinetic damage imparted by particles impacting an object while moving at only minute fractions of
c. Logically given the information presented, the energy required by the navigational deflectors to protect the ship while traveling at sublight speeds would be equivalent to that utilized during combat operations, and would explain the Enterprise's power expenditure shown protecting the transport at the beginning of "Mudd's Women". Now let's take it to a whole new level: we're not only going to have the vessel moving close to
c, we're going to take that value of distance traveled and multiply it by the appropriate integer value, and the problem becomes apparent, yes? If not, consider this associated article,
Starship Bumpers, which not only discusses the problem of the ISM in regard to relativistic velocities, it explains why the starship isn't just "shrugging that off without a problem." as you put it. Moving the ship into superluminal velocities would require the M/AM power output to be appreciably greater, as it's not just adding energy as you stated, it's in effect compounding per unit of time the amount of energy introduced into the system to be handled, and at some point, per what was shown in TOS, there is a mechanical limit to how much the system (M/AM reactor, navigational deflectors, bussard collectors, and the vessel itself) can handle before failure. Hence my observation about Scotty sweating bullets.
There are all kinds of flight sims and space sims where you can put some of these assumptions about speed and distance to the test. Celestia is a good one for this; pick a random direction and fly there, and you're unlikely to run into anything AT ALL, even if you have all the known asteroids and comets plotted. The same thing happens in Elite Dangerous, which does a pretty good idea of simulating the positions of asteroid belts and even the Jupiter Trojans in the Sol System; it turns out that, when cruising through an unfamiliar system, accidental encounters with planets are not only very rare, they're also totally avoidable with the smallest of course corrections, and are actually kind of difficult to pull off if you don't already have the planet/asteroid's orbital data in your computer.
Space is BIG. Planets, in the grand scheme of things, are tiny, tiny things.
Yes, space is big, but regardless, I would be curious if any of these sims take into account the effect that relativistic speeds have on the vessel's increase in mass. While chance encounters with massive objects may be a negligible concern, a vessel at moving at speed still has mass, and a vessel moving at relativistic speeds even more so. At warp the vessel has an indeterminate mass aspect (meaning we don't know how superluminal warp speeds effect the vessel's mass as it was never a relevant point of discussion in a story). It would be unfortunate if a vessel leaving orbit affected the orbital dynamics of things like the Earth-moon system or caused small asteroids to be yanked into new and unpredictable orbits. As such, it would probably still be a BAD IDEA to travel at high sublight speeds in system, not to even mention travel at warp. Just a point to consider.