If we think about the problem logically, we find that there's another aspect to the idea: if there were such a sheet of universal graphpaper, and that sheet did force physics to operate in such a way that we could identify an objects absolute motion relative to it, then that hypothetical sheet of graphpaper would (in a sense) have to exist within our universe, and the motion of bodies could once again be described using the principle of relativity, by treating our absolute frame as another (rather special) physical "thing". But it's perhaps slightly perverse to decide that the universe exactly one of these special things, with nothing else like it, so Occam's Razor pretty much demands that we reject the idea of a single absolute reference frame, unless there's compelling supporting evidence for it.
A more serious problem with the idea of an absolute, inviolable aetheric medium is that such a thing would appear to break some basic principles concerning cause and effect. Normally we assume that when a thing acts, it knows that it's acted ... that is, that there is a back-reaction for every reaction. We assume that if Object A exerts power over Object B, that A's ability to influence is somehow reduced, or at least altered in some way. There is no “something from nothing”, no expenditure of influence without a corresponding lessening of the bank account, and no free lunch. If A's ability to affect B was absolute and without consequence for A, then we could say that A's stock of influence appeared to be infinitely large. And if we are talking about an identifiable physical and quantifiable influence, it leads to some nasty mathematical results if we say that anything has an infinite quantity of a real physical thing. A further problem with these infinities is that they break accounting rules and the chain of causality. When asked where this influence comes from, we can't reverse the sequence of events and extrapolate any further back than the dictatorial rulings of our infinitely-strong metric, which then acts as a limit for any further logical analysis. It becomes a prior cause, a thing that can't be politely incorporated into a larger, fluid, mutually self-contained logical structure, but has its own separate anchor-point that doesnt relate to anthing else inside the structure, and allows it to dictate terms to everything else without retribution.
This sort of “absolute aether” is a way of saying that things simply happen in a certain way because they do, with no further analysis possible, and from a theoretical-analytical point of view, it's a dead end.It was partly Einstein's appreciation of this problem that led him to the conviction that spacetime itself had to be a stressable, flexible, malleable thing. The “medium” of Einstein's general theory was the background gravitational field (which also defined distances and times), but the assumed properties of this field were no longer absolute, but were affected by the properties of the physics that played out within it. Spacetime was an interactive, integrated part of physics. The “fabric of spacetime” deformed gravitomagnetically as objects passed through it, and spacetime itself was the medium by which masses communicated with and connected causally to other masses. There was an interplay between the properties of spacetime and the properties of matter and energy – as John Wheeler put it, “Matter tells space how to bend, space tells matter how to move”.
The more static, "fixed" spacetime of special relativity, Einstein later decided, was a somewhat distasteful creature. Certainly special relativity had done away with the idea of there being any absolute reference for location, and even for absolute independent values of distance and time, but the overall spacetime structure still had an “absolute” quality to it, in that the geometry of Minkowski spacetime was meant to control and define inertial physics, without its own properties being in any way affected (a slightly abstract version of "action without reaction"). Minkowski spacetime was still "absolute" in the geometrical sense.
... from the standpoint of the special theory of relativity we must say, continuum spatii et temporis est absolutum. In this latter statement absolutum means not only "physically real", but also "independent in its physical properties, having a physical effect, but not itself influenced by physical conditions".
It is contrary to the mode of thinking in science to conceive of a thing (the space-time continuum) which acts itself, but which cannot be acted upon. This is the reason why E. Mach was led to make the attempt to eliminate space as an active cause in the system of mechanics. According to him, a material particle does not move in unaccelerated motion relatively to space, but relatively to the centre of all the other masses in the universe; in this way the series of causes of mechanical phenomena was closed, in contrast to the mechanics of Newton and Galileo. In order to develop this idea within the limits of the modern theory of action through a medium, the properties of the space-time continuum which determine inertia must be regarded as field properties of space, analogous to the electromagnetic field.
... the gravitational field influences and even determines the metrical laws of the space-time continuum."
Because the word "relativity" is often equated with the predictions of specific theoretical implementations of the principle, it comes with a certain amount of historical baggage that isn't always useful when one wants to discuss a problem more generally. Sometimes it's more convenient to start from scratch and use a different form of words when trying to explain a relativistic principle without getting bogged down in historical implementational specifics. John Wheeler used the term "democratic principle" to refer to the idea that there's no single overriding cause that determines the forces on a particle, and another way of describing it might be to refer to the principle of mutuality, in that everything in the universe might be expected to not only have a vote in influencing anything that happens (subject to signal-propagation times), but also to be influenced itself in return.
So really, the principle of relativity in its broadest sense is just about going back to classical first principles: there's no action without origin and/or consequences, causality is A Good Idea, and nothing happens for no reason. These are somewhat pragmatic assumptions if we want to analyse the pattern of rules that the universe obeys – the first step is to assume that there IS a pattern.
There are, of course, more specific definitions of what the principle of relativity "says", which are tailored to the contexts of specific theories (usually Einstein's special and general theories). But we aren't obliged to use those existing definitions, and if we want a chance of discovering broader and deeper theories, we probably shouldn't.