No Stupid Questions

Insofar as USA law might apply, it may be useful for you to review the legal case involving Internet Archive's CDL program: https://en.wikipedia.org/wiki/Hachette_v._Internet_Archive

Since the realm of copyright law is inextricably tied to the question, I'm going to try to clarify some points. Firstly, "theft" has never been the correct legal analogy for copyright infringement. That misconception comes from a false equivalency in the late 20th Century to warn would-be infringers of the steep penalties; many Americans will remember the phrase "you wouldn't steal a car", even though the feds cannot charge copyright infringement as theft (which requires a tangible, non-duplicable item, like car theft or wage theft).

In the US at least, it's illegal to stream movies you don't own or don't have the license to stream.

Only the second part is correct: all copyrighted works are used per the license granted from the owner. Such a license may restrict the format that the work is delivered, but not always. The license that accompanies physical media is: 1) irrevocable, and 2) follows the disc's owner (recognized in USA law as the "doctrine of first sale"). So long as the disc is owned and intact, the license is good. Furthermore, under "fair use", it is allowed to make copies of works for either: a) time shifting (ie recording a live broadcast to watch it later) or b) to change the format, aka compatibility. The latter is why it's allowable to rip a DVD into a personal Jellyfin server. It's valid so long as the license is still good, which applies so long as you still own/possess the disc.

By participating in the co-op, when you stream a movie, ownership of that physical media and the digital copy is temporarily transferred to you.

Two counterexamples come to mind, the first being the Internet Archive case that I linked earlier. The second is a Supreme Court ruling against a company that rented miniature TV receivers located in metro areas across the country. In that case, SCOTUS found that although it's fine to rent out a TV receiver, the license for the over-the-air transmission was only valid within physical range of the signal. So conveying the TV content beyond the metro area created a copyright infringement, and the company was actively facilitating that. That company doesn't exist anymore, due to the crushing legal liability.

They are expensive for the library and don't have great selection

Most libraries are funded from a budget, and negotiate e-book and e-movie access based on an approximate estimate of concurrent users, not on a per-user basis. Otherwise, those libraries would have uncontrolled costs if everyone decides to stream Die Hard (1988) at the same time on Christmas Day; it's definitely a Christmas film. Quite frankly, most libraries would be thrilled if more people obtained library cards and used the services, because it justifies the budget for the library and proves its value to the community.

If you aren't finding the content you want at your library, the best thing to do is to request what you want. Libraries are always buying new materials or access to more services. But unless library cardholders voice an opinion, the librarians will just choose generically. Be the change you want to see.

Technologically, creating a co-op is always an possibility. But always remember that the very concept of a public library is "grandfathered" and if we had to reintroduce it, the establishment would never allow it. Cherish libraries as the crucial community resources that they are. The precise form might change, but the library role must always endure.

TL;DR: the idea is legally unsound. Instead, buy discs to form a community library and share the discs, basically a Blockbuster co-op. Or advocate for a better public library.