It's funny you should mention the First Amendment.
Florida, the leader of US book censorship, has been somewhat successful arguing in court that removing books because they have gay themes or characters is not a violation of First Amendment rights. This judge ruled that, because school libraries aren't "public forums", authors don't have the right to broadcast the "message" of their books through the library.
You'll often see this argument used by book banners: "it's not like we're saying the book can't exist! We're just saying it can't be in the library. If you want it so badly, go buy it yourself."
This is ridiculous for several reasons. 1) The point of a library is to serve its patrons by giving them, ALL of them, equitably, a wide range of materials to access. If books by and about BIPOC and queer people get removed en masse, as is happening in Florida, then clearly the library isn't serving all its patrons, which is discriminatory. 2) These are school libraries we're talking about. It's unlikely that these kids have the money and the means to purchase the books they want for themselves. And again, forcing those kids or their parents to shell out cash for books because their library isn't serving their needs is bigotry. Which is exactly the argument the ALA director uses in the above article.
How does this kind of shit affect what may happen with HR 7661?
Hard to say. Now, I am not a lawyer, but the bill as it currently stands is poorly written, purposely vague, and would be difficult to defend if it was challenged in court. But as we've seen in some of these book banning cases, some federal judges don't need a cogent argument to wave away the right to free speech.
The true insidiousness of 7661 is that it attempts to put pornography, which has a legal definition that this bill seems to ignore/rewrite, on the same standing as books that involve "gender dysphoria and transgenderism." Red states have already made huge in-roads in banning pornography despite the First Amendment, arguing that it's not AS protected as other forms of art or speech. Which had some legal precedent, right? Like, you can't walk into any store in the country and buy a glossy porn magazine. Sex shops that sell them are often restricted; they have to be so many yards from a school, for example. Pro-censorship orgs took those inches and got themselves miles in states like Florida and Georgia, where laws have essentially banned internet pornography.
Trans people will tell you that the last decade of anti-trans attacks has one goal: to remove us from public life, one way or another. If this bill were to pass, you'd suddenly have a legal question: if pornography and the mere portrayal of transgender people are conflated, and both are equally banned from school libraries, then.......shouldn't these trans books be banned from public libraries? From bookstores? From everywhere? The distribution of pornography is restricted, so why not these books? "These books are basically like pornography" turns into "These books ARE pornography!" And it doesn't take a genius to imagine all the terrible things that would follow.
(This is why I am so pro-pornography, but that's a whole 'nother post.)
This is all to say, your First Amendment rights are never guaranteed. Free speech is an idea, and an ideal, that requires a constant fight to maintain. That is why you need to call your representative today, and tomorrow, and the next day, to tell them you think this bill is trash. Raise hell now, because if they get this inch, they will take a mile.