I doubt that the battery pack has any problem at all with accepting charge at a measly 32 A, TBH. For a LR or P Model 3 that's only about 0.4 A per cell, which is tiny. The 2170 cells are rated at 4.6 Ah each, so 0.4 A is a C rate of only 0.087, nowhere near enough to start to warm the cells up.
The cells can easily charge at 2C or more, which is around 23 times higher than a 32 A AC charge.The delay between the charge point starting to advertise the availability of power and the charger(s) ramping up seems to be pretty variable, and probably related to temperature factors in components other than the battery pack. In all probability, the charger(s) ramp slowly to warm up their own components, to reduce the risk of damage from thermal shock.
The latter was a major problem during the development of the motor inverters for the Prius back in the 1990's. Toyota found that the IGBTs didn't like going from cold to high current operation, as the sudden heating at the microscopic level in the silicon caused fractures and device failures. The fix they came up with was to limit the current through the high power circuitry for a short time, so that the temperature rise was more gradual, over a period of tens of seconds, rather than a few µs.
I don't know for sure that this is what's happening, but it seems possible, especially as 32 A will be running two of the chargers at their maximum rating, of 16 A each. My guess is that the chargers can probably get pretty cold, my car has shown an internal temperature of down to -5°C this week, and, as the chargers are under the rear seat, my guess is that they were probably down around that temperature. When operating at full power the chargers probably run the switching devices at die temperatures around 50 to 60°C, maybe higher, and with the very small mass of the dies they will heat up very quickly if not limited in some way. Another clue is that I've noticed that our PV inverter behaves in exactly the same way. It's mounted on a north facing wall, outside, so can get pretty cold at night. We had some really bright sunshine yesterday morning, despite the cold temperature, and it took about 15 minutes for the inverter to ramp up to the maximum output. The giveaway was noticing that the DC input voltage was higher than it should be, a sure fire indication that the inverter is throttling the output power for some reason.