As a broad general rule, on a GPU each power input is independent. The PCIe slot power is connected to a power plane, and each PCIe connector goes to an independent 12V plane. From there, those 12V rails will run to different VRM phases. For simplicity ignoring the minor rails, say you have two memory phases and 12 core power phases. A GPU with a single 8 pin might run 1 memory phase and 2 core phases off the slot and 1 memory and 10 core off the 8 pin. This is fixed by the PCB design, the 12V rail from the slot would only run to 3 phases and the 12V from the power connector would run to the others. The GPU can't really draw more or less power dynamically from individual connectors, they usually increase in lockstep with rising draw.
It's also the reason that a card like my 3080 FTW3 draws very little from the slot even OCed (like 25W), and the 3rd PCIe plug always pulls less power than the first and second. That third plug is just physically connected to fewer power phases than one and two.
The 480 reference issue was that it was a 150W card with a single 6pin connector, so they split everything evenly so that at 150W it would pull 75W from the slot and 75W from the connector. 75W is actually a lot to pull from the slot though, and the 24pin ATX connector also only has 2 wires for all the 12V loads on the MB and in some cases that caused melty issues especially if the board pulled more than 150W and thus >75W from the slot. I'm not sure exactly what their fix ended up being, likely just disabling one of the phases on the slot side to adjust the balance, but it didn't really have anything to do with bad power bars or overloaded circuits.