What the starter does....
Before I say it, note one cathode is connected to the converted hot (from the ballast) and the other cathode is connected to the neutral...
Well see, when you flip the switch on...the starter CLOSES the circuit...making the electricity from ballast go through BOTH cathodes (filaments of the fluorescent lamp) from line, to one cathode, through starter then through 2nd cathode and out to neutral making the cathodes glow at start up...When it detects the gas ignited, then the starter disconnects. So when starter is disconnected during power on...the current goes through the tube instead of through the 2 cathodes and the starter. It only ignites when the gas is not carrying current.
At end of life, the cathodes (or one of them) loses emitter and somehow prevents the gas from glowing, so it kinda depletes the gas and it would attempt to turn on..but as soon as gas loses the current it turns off...making starter try to turn it back on...causing the strobing effect!
Makes sense eh?