I would blame the restriction on the speed at which objects can travel more on one of the two postulates that Einstein used to derive $E=mc^2$.
I wish Einstein's original derivation of $E=mc^2$ was taught in schools! It is such an amazing piece of work if you go through it in detail. But it's also cryptic, jumping very quickly through ideas that apparently seemed pretty "obvious" to Einstein, but which are far from that to the rest of us.
In any case, he derived it over the course of two papers. The first one defined the theory of special relativity, while the second and very short one derived his famous equation. It originally used $L$ for $E$, and Einstein never quite wrote it out the way we are used to seeing it.
His first paper began with with two simple postulates, which are:
(1) No test of mechanics or optics changes when you are moving without acceleration, and
(2) The speed of light is always constant when measured from such a moving frame.
Amazingly, that's all that is needed.
Now, if you want to point fingers at where exactly the idea that you cannot travel faster than light emerges from special relativity, I'd point at the second of Einstein's postulates: Every frame sees the same speed of light.
So why is that important? Picture it this way: If light must always travel at $c$ from your perspective, what happens if you launch a rocket capable of traveling at very nearly $c$, and your rocket in turn sends out a light pulse ahead of itself?
The rocket will see that pulse as traveling at $c$. However, as the person who launched the rocket, you must see something different, since otherwise the light beam emitted by the rocket would look like it's traveling at nearly $2c$, which would violate Einstein's second postulate.
So, the light pulse in front of the spaceship must necessarily travel at $c$ from your perspective also, and that in turn means that the spaceship must always remain behind any pulse of light that can be emitted. If you draw that out on paper, you get this quirky result that from your view, objects moving closer and closer to the speed of light must nonetheless always remain behind an actual beam of light, since any other result would enable you to see light a light pulse moving faster than $c$. Objects thus wind up getting "flattened" against the barrier represented by the speed of an actual light beam.
There are other consequences of this apparent flattening, which is called the Lorentz contraction, that I won't get into here. They include slowed time and increased mass, both of which can be derived from the original simple postulates that Einstein made.
So, the bottom line: It's more accurate to blame Einstein's assumed postulate of constant light speed for limiting material objects to traveling at sub-light speed, rather than blaming $E=mc^2$. And historically, Einstein didn't even derive the $E=mc^2$ result until his second addendum paper, which he published after he had already shown the other consequences of his postulates.
 Actually, there's an interesting minor secret buried in Einstein's postulates: One is missing. To ensure proper scaling of the results, you must add the following third postulate: If two groups of particles diverge from each other at speed $s$ along axis $x$, the orthogonal plane defined by the remaining two orthogonal axes $y$ and $z$ must remain invariant in scale between the two groups of particles. Or a lot less formally: $y$ and $z$ don't change, even though $x$ Lorentz contracts.
That point seems so obvious that it's usually either assumed or treated as an outcome of the other two postulates. However, you can't really derive it from the other two postulates since an infinite number of profiles that meet the first two postulates are possible if you allow variable scaling of the $yz$ plane. Lorentz had noticed this, but his thoughts about it were largely forgotten after Einstein's papers. In any case, when talking about Lorentz contraction of $x$ it makes sense to be explicit about the invariance (or lack thereof) of the remaining two spatial axes.