I really don't understand your point. Since ServiceWorkers provide an offline experience and can be inspected from the client, they should be better than no ServiceWorker with respect to privacy.
The main purposes of service workers are offline support and persistent background stuff. These are typically irrelevant in Private Browsing windows, which are designed for short-term use, and so service workers are almost always pure waste. (Not always, but almost always, and in line with the intended purpose and semantics of Private Browsing windows, though you can certainly use it in somewhat different ways.)
It’s similar to a large part of the reason why Local Storage and Indexed DB were historically disabled in Private Browsing: why store stuff when it’s only going to be deleted immediately?
Browsers have mostly headed in the direction of removing distinguishing features from their Private Browsing modes, partly to thwart detection of that mode, and partly because many things started to carelessly depend on the gated functionality. Firefox still doesn’t allow writing to Indexed DB (and only made Local Storage work far later than Chromium), and I’ve definitely encountered sites that just assumed it would work, and so fell over in early initialisation.
And so for this specific subthread: it’s ossification at work. If something is supposed to be able to change but never actually changes, you find when you try to change it that you can’t do so any more because people have built too much stuff that assumed it wouldn’t change.
I don't agree that private browsing is for short-term use. You can use a temporary storage for how long it is needed (in my case up until the operating system or the browser crashes, aka almost never).
Where can I find such design specification? All I can find is https://w3ctag.github.io/private-mode/ and https://www.w3.org/2001/tag/doc/private-browsing-modes/ that tell nothing about the short duration of a private session. Some emphasis is put on the necessity that private mode and normal mode should be transparent to the user (i.e. not breaking any normal feature). It should be not possible to tell the two modes apart. If you break a feature like ServiceWorker you are providing a way to detect a private browsing session, and that is a bad thing, worse than anything a SW could provide, in my opinion.
