I don't use "Save for Web" or "Export As", but this sounded like an interesting challenge so I did some tests.
I ran Save for Web on 3 different images from 3 different starting points of each image. The 3 starting points were:
1. ProPhoto and 16 bit.
2. sRGB and 16 bit
3. sRGB and 8 bit.
In SFW I used a quality of 80, Optimized, Embedded Color Profile, Convert to sRGB, and Metadata:All.
Suprisingly, my resulting jpeg files sizes were inconsistent. Sometimes the Prophoto 16bit starting point created a smaller jpeg. Sometimes not. That leaves only one other variable, and that is image content.
One of my images was a dull, flat landscape with lots of blank sky. The colors did not challenge the colorspace conversion. Although the image was converted from raw into Prophoto, the colors fit well within sRGB. The blank sky offered much opportunity for jpeg compression.
The other two images were "synthetic" images commonly used to test colorspace conversions and printing gamuts. One is called the "Granger Rainbow". The other is "Bill Atkinson's 28 balls". Google those names for more info. These 2 images, in Prophoto, contain RGB values that are way outside the sRGB colorspace.
When Save for Web (and Export As) is given a Prophoto 16bit image and told to make an sRGB 8bit jpeg it has to do three things.
1. Convert the colorspace
2. Convert the bit depth
3. Compress the values.
I think the key question here is in what sequence these three steps are done. I would think they are done in the sequence I've shown, but now I'm not sure. I challenge others participating in this thread to perform similar tests using images that do and do not cause significant changes in the colorspace conversion, and then help think this through.
Consider that converting from Prophoto to sRGB will cause some values to be moved closer together. Values that are closer together are more likely to get compressed. Bit depth conversion simply truncates the least significant bits. Some RGB values have nothing in those bits to begin with, so truncating them has no impact. In both cases, the jpeg compression will vary depending on when it is done and the range of starting values.