Commutating ferrite ring will saturate more quickly when load current increases...
The ring saturates, but because the frequency and mainly the current slope gets higher and amplitude increase the base drive (the voltage across the ring gets way higher, increasing the base drive way beyond what the transistor needs for the currents, so causes the Ts to go way higher), the quite long storage time in the transistors makes the current to go way larger. And this effect causes the current to go way beyond the normal operating current.
It is not uncommon to see currents 3x higher than normal, but more like 5x...
In reality the opposite is more of a problem: To keep the current from soaring really high, saturating the main inductor and frying the transistors, mainly when the lamp ignition fails (normally the high current rise is stopped by the discharge clamping the voltage across the resonant capacitor).
Also without this much of current boost, it won't be able to generate sufficient voltage for discharge ignition.
Taking an example with a 11W, 0.15A lamp, series inductor about 3mH, resonant capacitor 2.2nF, neglect the series coupling capacitor for the simplicity (it makes the things even worse, as it efectively reduces the inductance), fed from rectified 230V so about 300VDC so about 150VAC at the halfbridge. The 1'1t harmonic (important for the starting where it operates near the resonance) is almost the same, so ballpark assume it also the 150V (there are other simplifications here which make bigger errors)
Normal operating frequency would be about 36..40kHz, yielding the 0.15Arms (L/R time constant fed by a square wave).
Correction (messed up the quadratic root formula and not checking the result) and update:
A starting frequency around
140kHz 71kHz would yield about 0.45A and
220Vrms 450Vrms across the capacitor (Vstart = Ires * (1/(2*Pi*Freq); 150V/Ires = 2*Pi*Freq*L - 1/(2*Pi*Freq*C); assume operating on the inductive resonance side).
So although the original calculation was wrong, the 0.45..0.5A and 650..700Vpeak is what these selfoscillating ballasts do during startup (assume 2x MJE13001, 7turns on feedback, 2x 3turns for base drive T10 size core, about 22 Ohm of base resistors, 2.2 Ohm emitter resistors). They do not endure that for longer than few seconds, though, the transistors overheat.