While I agree that it's remarkable and likely not very helpful to so actively move to suppress these books, I think the argument that this matters for free speech is a poor one, deserving debunking.
Obviously there's the issue of protection of speech from government (protected by first amendment) vs. corporations (undermined by first amendment); but that's not the point I'm interested in.
There is a huge difference in speech that intentionally conveys an unpopular opinion for the sake of discussion, and speech in which that's not the point at all, and instead merely casually includes something toxic, or even accidentally. Essentially: whether the potentially unpopular or toxic message is explicit, vs. implicit.
The first category of speech (explicitly controversial) is highly important when it comes to transparency and the ability to think and debate clearly about uncomfortable issues. As far as I know, this is actually the only reason to protect speech in the first place; transparency and healthy debate matter.
The second category of speech (implicitly controversial) happens to be protected by the first amendment too; but it's more like collateral damage due to a limitations in the law (including because the distinction isn't always obvious).
Ideally, we'd replace the first amendment with something that more strongly protects the explicit category - even from corporate control - and at least tries to avoid protecting the second category, because such protections are easily abused (e.g. they're the reason why it's legal for a few corporations to effectively muzzle speech, and why it's so hard to control fake news).
As applied to dr. Seuss books - clearly there's no issue with transparency or debate here; no explicit point is being muzzled. There's no risk to free speech here, even if we happen to disagree on the policy of it's removal.
Put it this way: sometimes controlling speech (regardless of who is doing so) is harmful to free and open debate, hence it's worth protecting free speech. But conversely it does not hold that every infringement upon "free speech" necessarily impacts transparent and open debate. Quite the opposite; some forms of control can actually support transparency and open debate, e.g. control over spam and sheer volume (and this is one of the places where our current laws fail).
TL;DR: Slapping a "ZOMG free speech is under attack" statement under anything that happens to exercise editorial control misses the point of free speech in the first place. While we may have opinions on the specifics here, there is no question at all that this specific action does not harm the open, transparent debate needed for a healthy democracy. (And on a technical level it's alas the case that free speech is only protected from infringements by the state, not other powerful actors).