This is not correct. The practice of peer review didn't become established until the early 20th century. Einstein was famously indignant and withdrew his paper the first time a journal tried to subject him to it. Like many other journals, Nature didn't do peer review until the 01960s.
There are have of course always been journals whose editors published a proper subset of the letters they received, and we have a documented instance of this happening in the 17th century, though not on an ongoing basis.
> traditional academia...contributing less and less actual value
I could understand you writing this in 01822 but I cannot imagine what you are thinking writing it in 02022.
Traditional academia built the internet while industry was building X.25, Tymnet, CompuServe, and SONET (for the Advanced Intelligent Network, so the phone company could bill you more for more valuable information), and hobbyists were building FidoNet. Later, traditional academia built the WWW at CERN. Google and Yahoo were research projects at Stanford; Fecebutt was born at Harvard, albeit in the undergraduate space.
Microsoft developed the product that launched them by "stealing" computer time from Harvard to simulate the hardware it would run on.
AWS was built on Xen, which was created in traditional academia in the early 02000s before being spun out as a company.
So out of FAANG, FGA (plus Yahoo and Microsoft) are creatures of traditional academia; only AG are not. We could add the GNU project, informally hosted by the MIT AI lab from its inception.
So much for the internet; but maybe the internet is Bad Actually and we should not consider it "value contributed". What about larger issues in society, like pandemics, overpopulation, climate change, and energy?
Moderna created an mRNA vaccine for covid in March 02020, four months after the pandemic began and a few days after Chinese academics published the virus's genome; mRNA vaccine research was done entirely in traditional academia from 01989 to 02008, after which it was DARPA-funded for several more years.
The traditional academic research of Norman Borlaug tripled world food production in the 01970s; without it, civilization would have collapsed in famine.
Our knowledge of the looming climate catastrophe is entirely a product of traditional academia; and while many of the advances in photovoltaic cells essential to escape it have happened outside academia, many others have come from traditional academia, most recently PERC.
RISC-V was created by traditional academia. (So was Linux, and the Minix it was modeled on.)
John Goodenough, who invented lithium-ion batteries, is a professor at UT, though he was an Oxford don at the time, and it was Akira Yoshino in industry who brought it to market. (Yoshino is now a professor at Meijo University.)
> Teaching, which is supposed to be one of the core businesses of the academy,
There are people who suppose it to be one, but most of them are outside the academy. The core business of the academy is philosophy: the love of wisdom, the quest for truth. (Philosophy does, of course, predate the internet, so perhaps that's what you mean by "stuck in pre-internet ways of doing things".) Teaching is a means to that end; it pays the bills, gives philosophers valuable feedback on their ideas, and occasionally creates new philosophers.