If you seriously think the second principle goes without saying, then Galileo should be credited with discovering special relativity.
The second principle basically asserts that the laws of electromagnetism are physical laws valid in all frames, not just laws that hold in the frame of a medium. And since that was an actual view back then, it needed and needs to be said.
And really, by explicitly putting in the second postulate you can compute consequences about time and space relative to observers even if later we find out Maxwell needs to be changed, e.g. replaced with quantum electro dynamics.
You can even replace the word light in the second principle as long as there is an invariant speed and we call it $c$ then special relativity can make predictions even if it turns out there is no vacuum for things to move at a perfect speed $c$.
As a shorter answer: Special Relativity is a specific theory with its own specific predictions, theory about an invariant speed made to handle advances beyond Maxwell such as quantum electrodynamics without having to changes its own predictions.
"The second principle basically asserts that the laws of electromagnetism are physical laws valid in all frames, not just laws that hold in the frame of a medium. And since that was an actual view back then..." "an actual view back then" does not mean a law of physics back then. didn't Michaelson and Morley precede Einstein by 18 years?
Galilieo used the principle of relativity (the first principle) hundreds of years before Maxwell. Did Maxwell make SR? No, Maxwell wrote equations that people thought only held in a particular frame, the frame of the ether.
So what about Michaelson-Morley? Did they invent SR? No. They were trying the detect the motion of the Earth through the ether and failed to detect any such motion. Did this make people think that time was relative? No. It made people think the Earth was dragging the ether around with it.
Einstein made SR. And there were still other options. Some people were considering that most objects are made by balances of electromagnetic forces and that they could actually become smaller when moving relative to the frame were Maxwell holds. So the Lorentz contraction didn't have to be interpreted as about spacetime it could have been about objects moving relative to a fixed frame.
Which is an excellent way to teach relativity, using dodgeball.
In some fixed frame we could have some $t$, $x$, $y$, and $z$ and you could imagine a bunch of wearable devices that can internally record positions and times in that frame and each of the devices could compute $$v_s^2\Delta \tau^2=v_s^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2$$ for some fixed $v_s$ (the speed of a dodgeball in that frame, but it could be any fixed signalling speed such as the speed of sound in a medium atvrest in the fixed frame) and report the results $v_s^2\Delta \tau^2$=$v_s^2\Delta t^2$-$\Delta x^2$-$\Delta y^2$-$\Delta z^2.$ In particular they could report a running total of $\tau$ as recorded along the path they take in space and time.
Now they could use objects that travel at speed $v_s$ to communicate. For instance speed $v_s$ could be the speed of sound in a medium at rest in that one frame we fixed at the beginning, or they could all have the ability to throw dodgeballs at that speed. If they pass signals back and forth they could even define dodgeball time according to $\tau$ and dodgeball space according to $v_s$ times one half the round trip dodgeball time needed to send a signal there and back.
First you will notice that anyone at rest in the original frame has their dodgeball space and dodgeball time agree with actual space and time. But that anyone using dodgeball times ($\tau$s) and dodgeball distances ($v_s$ times one half the round trip travel dodgeball time of a signal) set by devices moving inertially to that frame will see strange effects like time dilation and length contraction.
You can get all the relativity of dodgeball simultaneity, dodgeball length contraction, dodgeball transformations. But there is still one frame where the dodgeballs move at speed $v_s$ but you do get a nice symmetry.
The nice symmetry is that everyone thinks the other frames record objects as having reduced dodgeball length. And that they can use the exact same transformations to go between any frames if they use the relative dodgeball velocity (dodgeball distance per dodgeball time of one inertial frame compared to the other).
At least for any frame that is inertial compared to the fixed frame and that moves at actual speed less than $v_s$ relative to that fixed frame. So you get a whole family of frames, each can transform to the other, each can use dodgeball time, each can use dodgeball distance, each has a kind of synchronization and a simultaneity that is frame specific, they see dodgeball length as a frame dependent thing. And they all agree when something moves at the speed of a dodge ball. Agree when they use nothing both dodgeball time and dodgeball distance.
But in this world the dodgeballs only move at actual speed $v_s$ in that one fixed frame. Everyone else is wrong in that there dodgeball time is not time and there dodgeball distance is not distance and they have a flawed notion of simultaneity based on passing dodgeballs back and forth.
But if they only use the dodgeball time and the dodgeball distance then none of them can tell which is the one in the fixed inertial frame.
And that is truly where relativity comes in. First you establish that there is a different kind of symmetry, one that allows an invariant speed. In the limit where the speed is infinite you get Galilean relativity. And the speed could be anything, it could be the speed of light or the speed of sound in a medium where sound is fast or where sound is slow.
Then you argue that any frame is just as good as the true inertial frame. And that's when you get a SR (for any invariant velocity), a theory with relative simultaneity, length contraction, time dilation, a group of transformations that hold a speed $v_s$ invariant.
And when you pick the particular theory with $v_s=c$ then you get Einstein's SR. And you get it even if there is photon-photon scattering and Maxwell is wrong. You get it even if photons are massive and Maxwell is wrong. You get it even if there never is a true and perfect vacuum. You get it by just having a fixed invariant finite speed.
You get it by having a different group of transformations. But that didn't tell you that space and time were weird, the transformations for dodgeball were just for dodgeball time and dodgeball space.
Being open to those transformations is key. And using the first principle of Relativity isn't enough. The fact that no frame feels special is key, but the goal is to have a specific theory that fixes a particular speed. Not to have a bunch of different theories for each possible law of physics.
Because otherwise the principle of relativity is meaningless.
You could have a theory where light travels at speed $c$ in just one frame using actual time and actual distance. And then you could say that everyone measures dodgeball time and dodgeball distance with their clocks and decays and their rulers and such. And that they wrongly think light moves at speed $c$ in their frame, it only moves at an invariant dodgeball speed (dodgeball distance per dodgeball time).
Such a theory has each frame feel equally good (none of them can tell, using dodgeball time and dodgeball distance, their velocity relative to the frame where Maxwell holds) so it doesn't violate the first principle. But it is still not truly Copernican because some one (but unknown) frame has their dodgeball time agree with actual time and their dodgeball distance agree with actual distance.
The point isn't to just have Maxwell describe a medium it is to have an invariant speed, and specifically have that speed be $c$.
That's why Einstein didn't require that every single bit of Maxwell hold, just that there be a speed that everyone agrees on. If for instance a photon has nonzero mass then $c$ isn't the speed any individual light beam travels at (and it isn't anyway since there isn't a perfect vacuum) it is a limiting and invariant speed.
The whole point is that SR is a complete theory predicting an invariant speed $c$ and it works for QED (where Maxwell isn't true) and it works for the real world (where there is never a perfect vacuum). We can use SR in places like QED where Maxwell is wrong.
It's saying that if you fix an invariant speed then people won't agree on the other things like simultaneity, and if you make that invariant speed equal to $c$ then laws like Maxwell can be equations for any inertial frame instead of just one frame.