Does the first postulate of special relativity imply constant speed of light?

The first does not imply the second, but you are nevertheless on the track of something which I will try to spell out.

The second postulate can be made in a more minimal way. Instead of mentioning all inertial frames, one need only mention one:

  1. There exists an inertial frame in which the speed of light in vacuum is independent of the motion of the source.

One can then bring in the first postulate and argue that if there is just one inertial frame with this property, then it must also be true in all inertial frames. One can also make a further argument that the speed observed for light in vacuum must be the same in all frames (as long as the term 'vacuum' is not being used to refer to some medium which itself could have detectable motion). So your suspicion was almost right: the first postulate does have something to say about the universality of the speed of light. But nevertheless the principle of relativity (i.e. postulate 1) on its own cannot be used to derive anything about the speed of light nor the maximum speed for signals. For, don't forget, Newtonian physics respects the principle of relativity (your postulate number 1), and in Newtonian (or Galilean) physics there is no finite maximum speed. In the Galilean version of spacetime, signals can in principle go infinitely fast, and everyone agrees about which events are simultaneous and which are not. Therefore the second postulate certainly does not entirely follow from the first.

Finally, I would like to note that you do not need to mention light when stating the second postulate, and arguably it is useful not to bring it in. It suffices to say that there is a finite maximum speed at which causation can happen, and leave it at that. By "causation can happen" I mean there is a finite maximum speed at which an influence from any given event can carry effects to other events. Sometimes we use the word "signals" for such influences; then the statement would be that there is a finite maximum speed for signals. Or, to hone it down a little more:

  1. There exists an inertial frame in which there is a finite maximum speed for signals.

It's pointless to worry about whether the two postulates are logically independent. Einstein's 1905 axiomatization of special relativity was not a set of axioms of the kind that mathematicians have in mind when they talk about a formal theory.

Einstein wanted to take Maxwell's equations as fundamental laws of physics, discard the aether, and apply the same symmetry principle (what we would today call Lorentz invariance) to both mechanics and optics.

His contemporaries would have thought of Maxwell's equations as applying only in the rest frame of the aether, and they did not agree on what overarching framework, if any, should be shared by mechanics and optics.

If you assume that the speed of light should be a quantity that is predicted by the laws of physics, then postulate 2 follows from postulate 1. Einstein's contemporaties did not share this assumption.