I dont think there is any emitter coating on the probe - For the short time it takes to start the main arc who cares if it rectifies at the moment of starting ?
Not the probe, but when the main electrode lack the emission coating it is designed for, as a consequence it runs way hotter than the lamp is designed for.
And the problem would be then the radiated heat from the main electrode, melting the nearby starting probe not designed for such heat it receives.
For the HX autotransformer and crest factor: There is one fault, which may lead to such problem: Magnetic shunt being thinner than designed (missing plates in the stack inserted between the windings).
I don't know, how exactly the ballasts are made to ensure the correct final current, but if the process involves "tunning" of the load current by inserting more/fewer of the plates into that stack (to compensate for the gap size variation), once they pick accidentally too long plates (so it left narrower gap; just wrong gauge, e.g. intended for other ballast model on the same base core size), they may reach the rated final current on the adjustment jig by just inserting thinner stack.
So although the final current at the rated voltage would seem to be OK, mainly during the lamp warmup the shunt will heavily saturate and cause high current crest factor.
If the ballast is tested using only single (typical rated hot lamp) load simulator (you can not use real lamps for ballast production tests, at first it won't survive longer than a week and seconds, it's parameters would vary way too much) and single fixed voltage, such test is not able to detect that error.