How much power can a flyback transformer core transform?

I have a bunch of dead TV line transformers that I salvaged the ferrite core from. Typically these run around 20 to 40kHz or so.

If I remove the air gap and use these cores as traditional high frequency transformers with a primary, secondary, and maybe auxiliary feedback in an Resonant LLC or forward converter, how much power could be pushed through? My guess would be up to 1000W given the size compared to similar transformers in PC power supplies.

Would the answer depend on switching frequency? I'd imagine saturation and hysteresis/eddy currents would be an important consideration. Would such a core design (being just 2 C sections) would have a non-ideal 'K' coefficient and some leakage inductance. For an LC or LLC converter that's probably desirable.

My intent is to play around with creating a 500W to 1000W inverter. converting maybe 48V to 120V or even 3 phase.(or whatever I can get out of it, lol