I think you’re overestimating the importance of limb independence. Playing complicated polyrhythms is not a common drumming task, and before you get there you’ll want to learn complicated monorhythms. Playing a solid groove does not require independence as much as it requires coordination.
I would learn limb coordination first. Luckily, that’s exactly what rudiments can teach you. The trick is, don’t just play rudiments on the snare or practice pad. Learn to play them all over the kit and even with your feet.
The other important category for beginning drummers is technique. Your grips are crucial, and how to use grips and how to change grips (e.g., during a roll) is crucial. This is one area where the rudiments help with diagnosing and working the fundamentals. As you try to speed up (close) each rudiments, your fundamental technique will have to be up to the task.
How to play the kick and how to hit cymbals properly are two other basic beginner techniques that would be good to master ASAP.
Finally I would say ergonomics is the third big category. This is closely tied with techniques because the proper techniques should be ergonomic as well as good sounding. How to sit, height of throne, arrangement of the kit, reaching versus not, how to use your arm to control dynamics, etc.
All of those are more fundamental than limb independence. Again, what you really want is for your different limbs to be coordinated, so the action of one can trigger the action of another. You don’t want them all playing separate rhythms (except in the rare cases where you do, but beginners don’t need this). If you did achieve perfect independence and used it to play like a funk rhythm, well then you would sound just like a drum machine. The natural flow and coordination between hands and feet is what creates the musical feel of each rhythm. For that to happen, the limbs have to work together, not separately.
Overall strategy I recommend learning rudiments and songs. Find music you like and listen for the simplest beats in that genre or by that band, etc. Allow practice to include playing, in the playful sense. I mean if it isn’t partly enjoyable why do all the work?
PS: get a teacher!