Have some universities already tried to temporarily disable their journal subscriptions
This has certainly happened, at large scale to boot. For example in 2014 60+ German institutions all let their contracts with Elsevier expire while they negotiated a new contract. Elsevier went ahead and gave them free access while negotiations were ongoing. In 2018-2019, Norway and Sweden did the same thing at national level, and so did the University of California.

If you want to know more about how this impacted usage, I'm sure you can ask your institution librarian for details. I'm sure they can tell you a lot: how they decide which journals to subscribe to, what would happen (e.g. how many complaints) if they unsubscribed from a popular journal, how they can secure access even if they unsubscribed, and so on.