https://w3c.github.io/csswg-drafts/css-content/#content-property doesn't define whether the content it generates is copied when a selection containing it is copied, and Chrome 106 and Firefox 102 don't copy the generated content. For quotes, it's important that the generated content gets copied, although there are probably other use cases where it's important that it not get copied. So I suspect there should be a new property to generate copyable content, rather than changing the behavior of the content: property.
See also #6222 for the question of searching for generated content.