When it comes to filling phenolic tubing (at least the PML stuff i`ve used) and sometimes BlueTube ,those spiral grooves can be pretty deep ,in a relative way.
For me,it`s Bondo Red Spot putty in a tube ,although I have heard the 3M Acryl Blue is wonderful stuff also ,I just can`t seem to buy it at local shops (may need an automotive supply store)
Using an overspray of primer to fill these grooves would not be very effective at all ,and must be filled prior to prime/paint.
I remember years back ,I used a light ,fluffy looking "spackle" type filler on my first PML Phenolic project ,and it worked very well and was water clean up with minimal shrinkage.
On some cardboard tubes ,the spirals can be very minimal ,so a watered down woodfiller or as Micromeister mentioned ,several coats of high solids primer may be all that`s needed.
But alas ,I`m stuck on spot putty now and have been for a few years.It`s easy to apply ,dries fast and has minimal shrinkage when applied properly and it sands very very easily.If you use a good quality sandpaper ,it won`t clog too badly ,just check the paper once in awhile and tap out the dust often.
Paul T