Students built a 16th-century engineer’s book-reading machine

Along these lines it somewhat reminds me of the iBooks shelf from pre iOS7 days

Books were bigger back then, so imagine trying to create a lazy susan to hold 8 books that were ~2 feet wide when open. You'd need a circle more than 10 feet in diameter.

There's generally unaffiliated access. If you want to take things out on loan that's an annual fee but at least at the UCs it's pretty permissable. I've even gotten access to rare book collections, just as random joe q public.

There's usually a public access backdoor somewhere. Almost nobody uses it but it's there.

Usually I'll find maybe a rare copy of something at some scholarly library, they've always been ok with me if I email ahead. It's been really painless.

Some libraries even have digitisation devices you can use as a replacement to the old copy machines.

That's more than enough for me, I don't actually need to leave with the library's copy. Some photos are fine.

Apparently even the restrictive Caltech will do it although I haven't yet had a need to exercise their system

This is just categorically untrue. I've been able to peruse and yes even read very rare manuscripts at the NYU library all while being a mere "civilian" (no uni affiliation)

I'm sure the 16th century engineer realized that too and designed it more because it was an amusing idea than anything practical. There's a reason it wasn't built until now.

Think standing desk, you can raise or lower the books to your ideal height. Further, why stop at one? You could have a semicircle of 4 of these very easily. That said, this didn’t catch on for obvious reasons.

I don't think that really works with large leather bound books, especially in low light.

Three rows of three [vellum] books on a table: You're looking at lifting a several kilogram book at a stretch of 60-80cm (quartos are about 30cm tall) ... my reach is only ~60cm (at ~175cm of height). I think what you're proposing is far less convenient than you're imagining. Especially working from candlelight (in UK most people would have used animal fat lamps at this time, I gather; but if you could afford 8 books I'm guessing you could afford proper candles).

One possible reason it might have seemed slightly more useful in its time is that many books much bigger than what we expect today. I've seen some books from the 13th century that are over 20 inches across when open and quite heavy and I don't think those books were particularly large. (Codex Gigas is over 40 inches across when opened and weights 150 pounds, but is probably the largest book we know of.) Having 8 books of 24 inches across open next to each other would take 16 feet or so of linear space. In the drawings it doesn't look big enough to handle these large books unless it was made slightly wider.

Of course your objections are probably very much why it was never built until now. :)