I figured I'd get around to looking at the supply on a scope at some point, but you're saying that the ADC itself isn't very good?
Four times as many samples => half the noise (given some reasonable assumptions).
Personally, I would have really liked to get higher sampling frequency on the ADC. The raspis are quite limited in that regard compared to more "typical" microcontrollers (would expect 1MSPS or more). Supposedly some people had success overclocking the whole peripheral by a factor of 8, but I'm not so sure that is a good idea...
n The RP2040 ADC has a DNL that is mostly flat, and below 1 LSB. However at four values — 512, 1,536, 2,560, and 3,584 — the ADC’s DNL error peaks above this value. The ENOB for the ADC has been reduced from 9-bits (simulated) to 8.7-bits (measured), see Section 4.9.3. The DNL errors will somewhat limit the performance of the ADC dependent on use case.
For the record, I love these chips, despite the problems that came up.
There is also some noise from the Pi Pico power supply which could be a good thing if you're willing to average or filter over a large number of samples.
More details here: https://pico-adc.markomo.me/