The adapter is passive so it is the car and charger that normally limit the current and total power used during charging.
Tesla cars will happily pull 500A at low states of charge with proper pre-conditioning which is the peak limit for almost all existing CCS chargers (one CCS model can now apparently put out 540A).
The new 250 kW is probably just meant to imply 500A x 500V. The kW figure isn’t really relevant to the operating conditions of the adapter itself, only the ~500A current is at issue since that is what generates heat during charging and all existing Teslas charge at under 500V.
I have used my Tesla Korea OEM adapter, with its putative 300A rating, to charge at 350 kW labeled chargers that pulled 500A and reached ~184 kW on my Model Y LR even at ~100F on sun-baked afternoons in Nevada. After a minute or two of charging at around 180 kW, the power dropped to near 100 kW for awhile and then returned to around 150 kW before settling into the usual Model Y Supercharger charging curve. This strongly implies that the charger and/or the car measured excess heat at the adapter and temporarily reduced the charging current. Such thermocouples are widely thought to exist in the CCS plug and in the Tesla charging inlet to allow the measurement of temperatures at the charge pins.
Any thermal protections built into the adapter itself would work by entirely disconnecting the circuit (perhaps permanently with a fuse) and so never came into play. The adapter protection would be there as a fallback to prevent catastrophic failures.
If future CCS chargers begin to support output well above 540A or Tesla begins to deliver cars that charge well above 500V then the car itself might have to limit CCS charging to 500V and 500A to avoid exceeding the real capabilities of the passive adapters that people are using.
