I remember also having the intuition that copyright should be seen as a free speech violation, and the existence of a major tension between them should be obvious. Although the Supreme Court said in Harper and Row v. Nation that copyright is meant to be "the engine of free expression", and publishers have
really enjoyed that phrase.
When I was younger and working on some of this stuff I was struck by the hyperbole about incentives. It seemed obvious to me that people love engaging in cultural expression and are very eager to do so. So the idea that the government needs to look after how much money they can make seemed pretty ridiculous, and especially when people would imply that "no one" would write books, poems, software, etc., or "no one" would take photos, without exclusivity over the use of those works. But of course, there was lots of cultural production long before copyright, including works that we really appreciate and admire.
Basically, the copyright subsidy is trying to promote forms of professionalization of cultural production, where more people will be able to do it for a living in a focused way, and more people will be able to do it in a way that requires significant monetary investment. My most concrete experience of this was when I was on a Hollywood studio tour (organized as part of a meeting literally related to copyright policy) and the movie people showed off an incredibly expensive set and said that strict copyright enforcement was what allowed them to make movies that required such elaborate sets.
And that seems essentially right: of course people will make movies without copyright exclusivity, and of course they could make money from their creative work in other ways, but it would be much harder for them to find huge investments to create particular resource-intensive works. (Ultimately that could also include time-intensive individual productions by people who just don't have enough free time.)
It was never clear to me who decided, or how, that the expensive Hollywood thing was worth the downsides in lost freedom, but it's also clear that lots of people, inside and outside the film industry, are very enthusiastic about it and grateful for it.
And we do get appreciably more professional cultural production as a result of copyright enforcement, even if it's transparently silly to say that we wouldn't have culture or people wouldn't do creative work without it. But it obviously continues to feel like a very crude instrument to me, and I'm sad that potential alternative mechanisms continue to get relatively little attention.
Patents are different but have some similar stories: many purported inventions aren't genuinely original (partly because it's also hard to incentivize people to document "the state of the art" for defensive purposes) but there are some areas where some kinds of technological development are really capital-intensive, and patents do provide a mechanism to incentivize investors to make heavy capital investments in those (because the exclusivity provides a clearer path for them to potentially make large financial returns to those capital investment).
I still agree with the general point that these systems are working by trading off people's freedom (to make copies, to use their technical knowledge) and that they have a feeling of bluntness and crudeness to them. Partly the government doesn't want to get involved in deciding in a case-by-case way what would be "necessary" or whether copyrights or patents were harmful or beneficial in a particular setting, so they want to have uniform rules where lots of people easily qualify for copyrights and patents (although the specific legal standards for subject matter eligibility are rather different between the two).
There's some solid economic theory in favor of copyrights and patents, too, in terms of underproduction of public goods and internalization of externalities. But it's somehow not that common that they get discussed mainly in those terms.