Sorry for the delayed reply. This deserved a keyboard and monitor. My previous replies have all been typed out on a phone. Brace yourself for a wall of text!

I’m going to reorder your reply a bit here, so bear with my. No words or sentences have been changed, just their order.

Fundamentals first:

Here’s what makes sense to me after your explanation: the buckets have the lid off all the time, so anytime light falls on them, it gets collected. […] An assumption I make in this case is that the buckets don’t really have a lid at all, you don’t control when they close, you can only control how much light enters by means of the shutter. Is this assumption correct?

This is correct. The steps remain:

• Empty the bucket you want to measure
• Allow it to fill for the appropriate amount of time (shutter duration)
• Measure how full the bucket is

A mechanical shutter can provide a lid, but there is not a lid in the case of an electronic shutter.

This makes perfect sense in mechanical shutter: they are there in complete darkness waiting for light, and when the curtains fly over them some light falls inside the buckets, and this light all fell almost at the same time. Basically they all saw the same thing. After that they stay open, but since it’s total darkness the amount of light stays the same, and the readout can calmly go to each bucket and measure what’s inside. This works perfectly.

The key bit of this paragraph is almost. Although 1/250 is pretty darn fast, you can experience rolling shutter with mechanical shutters. Here’s another example. Look at the ball touching the bat in the top of the photo and the shadow of the ball moving away from the bottom. To save you two clicks:

The below was taken on a Canon EOS-1D X, which is a DSLR

Not sure what the photos below were taken on, but it was with the mechanical shutter active

Now, I need to apply the same logic to the electronic shutter because otherwise I go crazy. In the electronic shutter you must have a way to control that lid otherwise you can’t control the exposure. In the mechanical shutter the curtains did the job, in the electronic one light is already falling inside those buckets all the time so you must have a way to limit that, and it cannot be the readout alone because you would be limited to the readout speed as the shortest exposure time, which is not the case.

I think this might be where you’re getting hung up: there is no lid for the buckets when using an electronic shutter. As the electronic shutter sweeps the frame from top to bottom it’s emptying buckets as they’re reached, pausing, allowing them to fill for the exposure duration, measuring how full the bucket is, and then moving on to the next bucket. This is the reason why rolling shutter is so much more pronounced on electronic shutter cameras with slow readout speeds - it has to wait for the prior bucket to be read before emptying the next one and starting the process all over.

To put this in pseudo code for a mechanical shutter:

empty all buckets  // fast
sweep mechanical shutter across lens  // 1/250 second in total
for each bucket in buckets: read the value // no light hits any bucket during this process

A nd an electronic shutter

for each bucket in buckets:
empty bucket // fast
allow bucket to fill for exposure duration // usually pretty fast, but note that you're going to be doing this a lot
read how full the bucket is // slow and you're also going to be doing this a lot

With a way to limit the amount of light then it would make sense that, in ES, if I imagine the readout as the thing that adds pieces of the picture sequentially, each bucket waits with its lid closed, it opens the lid for a tiny bit,

With an ES there is no way to close the bucket, but you can empty them on demand

it’s filled by the set amount of light, and is immediately measured by the readout and a piece is added to the picture, but since it takes a long time to measure this one bucket,

This is accurate! You’re very close

the next one just stands waiting with its lid closed.

The next one is just hanging out filling (or maybe even saturating), but it doesn’t matter because it’s going to get emptied before it gets measured.

For this reason the moment of the collection of light is limited to the readout speed, and it makes sense that by the time you’re done with all buckets a lot of time has passed because the readout is slow, so the rolling shutter effect happens

This is also accurate! But also: rolling shutter can totally happen with mechanical shutters too. The only way to truly eliminate it is by using a global shutter. If you want to see one in action, the only video I can vouch for is Chris and Jordan’s first impressions video on the A9III. It’s very weird seeing a lot of high speed sports with zero distortion.

but they all had their lid lifted for the set amount of time so the amount of light is correct

I hope you’ve got it by this point, but the way we control the buckets is by dumping on demand and then measuring right after their exposure period has elapsed.