I see it as less about signaling and more about proof.

A long time ago, a high school diploma was rigorous enough that people actually failed. A significant number of people did not earn one. Now, I don't even know of a friend of a friend who has failed to earn one, even if they just kept on retaking the courses at relatively easy schools until they passed.

It isn't clear to me what holding a high school diploma indicates about a person anymore simply because nearly everyone gets one.

What does holding a high school diploma prove?

This is what the "signaling arms race" refers to.

The other model, human capital theory, says that a high school diploma proves that the student has acquired specific skills (algebra, European history, etc) and therefore built up human capital. Signaling theory says the specific skills are not the point.

Signaling is synonymous with social proof in this context.

Funding education through employer taxes based on the education requirements they desire was an idea that I heard that might have some merit. You want someone with a bachelors? You pay X amount per year for that employee. A masters? Even more. You expose the employer to the cost of their desire, and thereby suss out if it is a want or a genuine need. Very similar to carbon taxes.

If the tax is more expensive than on the job training, employees will get on the job training, possibly through an apprenticeship.

Would that alter employer behavior or just job descriptions?

They could just set their employer talent management system to filter out applicants without those degrees.

If you employ them, and they have the credential, you still pay the tax regardless of job req contents.

The Elephant In The Brain has a chapter on education which clearly draws a lot from Bryan's book The Case Against Education. Interestingly Bryan Caplan was one of book's pre-publishing reviewers.

Education should also have the task to form the adolescent, to make them fit for life, i. e. resilient and happy. Signaling might become less and less successful and neccessary as decades pass.