Simple question. Guy writes a Star Trek episode. In the old days, he gets a check in the mail every time it airs. Should he get a check every time it’s streamed? How does that happen without opening the books?
That isn't even something they are asking for, so it's entirely a theoretical question. You are basically making up a metric that doesn't currently exist. If they did negotiate for some sort of pay-per-play deals, sure, they would have to make the numbers available.
If they suddenly released the numbers, it would not change any of that. Basically, streaming works are for-hire, they hire you for a job, and, knowing that there are no residuals in streaming, that is part of your manager/agents job to make sure that the pay they negotiate for takes that into account. (And, in a lot of cases, they actually do better on front than the .50 cent residual check years later.)
But again, that's not the reason that they are asking for the "books" of these private corporations to be available - and it's not the writers that are demanding it, it's the actors, for the reasons I stated above. It has zero to do with residuals, it's so they have a number on their resumes that they can use as bargaining chips when negotiating between companies for higher salaries ("I got $5 million for my last picture from Netflix, and got 1 billion minutes of views, so I deserve $7 million this time from you").
FWIW, both sides do have legitimate complaints, though it's really the writers who are most shafted. In both cases, it's because the business model is in flux and is in its true infancy - it's like it's 1960 and we are talking about television here, as I believe you pointed out. The thing is, people got very used to the old employment model, but that was based on most content being either big studio movie productions, or television shows on commercial ad-based broadcasts that did 26 episodes a year. The latter especially, is not coming back.
The main issue for the writers is not residuals but actually the "mini-room" phenomenon, which George RR Martin goes into great detail about on his
(not a)Blog. It's actually a very legitimate gripe - and among creatives, it's why the writers really have been getting the raw hand of the deal as everyone is figuring out how this business that has operated the same way for more than a half a century gets reworked. Writing in Hollywood for television used to pretty much be like a structured day-job, and it just doesn't work that way when it's not a network producing 26 episodes and working over 10 months a year.
That's why it's a shame that the actors have jumped in and taken the spotlight (who would have thought!), because the writers are the ones who have really been hurting, and taking away the focus on that has really caused an erosion of public interest/support.