Not so crucial, actually. Well, to the transmitter parts...but the actual valuable stuff Vanguard I did (and continues to do!) involves optically tracking it to measure the density and behavior of the outer atmosphere. That doesn't take anything on board at all--all the work is done of the ground. You just need to fix the satellite's orbit after launch and take periodic measurements afterwards.
It would probably be heavier and bigger for purposes of practicality, but a first satellite barely has to do anything, "to prove it was in stable orbit" or not. Any competent ground observer will quickly notice that something very odd and new has shown up within weeks, if not days of the launch, and a relatively small bit of math (of the sort routinely used by astronomers to compute the trajectories of comets and asteroids) will show that it's in Earth orbit. Of course, it helps if you have Baker-Nunn cameras and Operation Moonwatch--but they aren't necessary. Just a simple metal sphere--Vanguard with no instruments or Sputnik I with no radio--is plenty to demonstrate the principle. And, as I mentioned, can do useful scientific work with no further modifications.
Again, there are lots of useful things that passive satellites can do (for a certain definition of "useful"). As I mentioned, help measure the properties of the upper atmosphere; serve as communications relays for radio signals (a la Echo I and II); or serve as navigational aids, literally artificial Moons (for celestial navigation). None of these require any on-board instruments and indeed require virtually no on-board equipment, certainly nothing beyond the capability of the early 1950s or late 1940s.