To explain further, I have loved books for a very long time. I've always been interested in how they're made, and the possible "illnesses" that can befall books and how one goes about to "cure" them. I want to do it as a job, and the research I've done has lead me to learn that there's printing press bookbinding, with machines, and then "craft" bookbinding, the work by hand. I want to know how I can obtain the skills to become a "craft" bookbinder. The "old" way. Are such programs offered at US universities? If not, what is the usual path for gaining such skills in the US?
There are two main academic tracks I'm aware of that care about craft bookbinding, in a way that is beyond hobby: fine arts, and "library and archives".
In schools of fine arts it is not uncommon to have "3D design" classes that can include craft binding concepts and techniques as an option. Other classes can focus on leather-work, print-making, binding, embossing and imprinting (often used with book covers), etc. Depending on the size of the school and the local area, they may have occasional seminars specific to book-binding, and some will even be open to non-degree seeking students. So if you have a local arts program, it may be worth a visit/information/tour, and you can ask them if they may have any programming or faculty with experience in your area of interest. In may indeed just be a single person and you could try to take a class with them, or they may even be willing to work with you directly in a helper/apprentice/tutor role. You might also be interested in fine art restoration and preservation, though this is a not a program universally available at all art schools.
Artists are the primary 'consumers' and practitioners of modern craft book binding, because they care about one-offs and details in a way that commercial printers just don't (and can't) care about in the same way. Some commercial shops, especially serving high-end marketing/promotional companies do still buy craft binding work, but it would be hard for someone just with an interest in the practice to connect through this route.
The other main academic track is in library and archives, specifically in the sub-areas of preservation and restoration. This usually isn't focused only on books, though there are book-preservation specialties - but most people in the area will study a broad array of document preservation, from arts to historical documents to books and other artifacts. This will get you much closer to book illnesses if there is also a need for restoration. This is more of an academic path, as I don't know that they are as open to non-degree seeking students who just want to learn as book binding - but it might be a lead you can try to follow.
There may also be hobby-focused classes offered at community/trade schools in your area, as Buffy mentioned - which are also definitely worth a check, and probably your first point of contact if you are mostly interested specifically in learning how just to work with books.
It is possible that there are some academic programs. They would be more likely at a Community College, I think. But you might be able to obtain an internship or apprenticeship at some custom publisher or even a major book publisher.
To find such an opportunity I'd suggest the following steps. Your local library has a need to re-bind books and knows who does that. Find out the contact information for such a service and, starting with them, work up the chain of contacts to the people who do the actual production.
Other than librarians, you could also talk to a book-rep who visits your school or at an academic conference. Again, work up the chain of contacts to the production department.
There are a number of presses in the US who do custom small editions for authors and others. Thorndike Press in Maine used to be one of those, but I don't know if they still do it. But there are others. Search for Custom Publishing. But they may use suppliers of services and you will need to work to find who actually does the work. You are looking for, and may find, a company that actually uses hand crafting to bind books and even those that design the bindings using non-standard materials.
If you can find a really high quality book production outfit, you might explore either an internship or an apprenticeship with one of them. It might be paid or not. But even an entry level regular position will give you an opportunity to learn the skills.
Another source, but harder to find, are individual artisans who design and manufacture their own books (poetry, art, ...). I met one within the past year at a craft fair. But a search for "hand made books" might turn up something.
I was taught craft bookbinding in a university town - by the bookbinder to the university library. But he did not award degrees. Do you want to learn the craft or do you want a degree certificate? The two wishes may not be compatible.
My spouse worked in a small bookbinding shop in her university library as an undergrad as a work study assignment. The shop was there to rebind and do other preservation work on the library's older books. It was very much a matter of learning under the supervision of one of the full-time employees.
The closest program would probably be a Library Science degree (e.g. MLS, MLIS). The actual percentage of the program that would directly deal with physical book construction and repair is almost certainly quite low, but what you will likely encounter in these programs are fellow enthusiasts who share your love of books and are likely to be able to point you toward potential opportunities.