I think you need to re-read that law yourself. Section 230 holds Internet platforms not liable for illegal or libelous speech because they aren't the publisher of that speech. If you read on it also says explicitly they can't be held accountable for:

"any action voluntarily taken in good faith to restrict access to or availability of material that the provider or user considers to be obscene, lewd, lascivious, filthy, excessively violent, harassing, or otherwise objectionable, whether or not such material is constitutionally protected"

Any Internet platform can moderate to their heart's content, even if it goes beyond the extent demanded by the law. There are no constitutional free speech protections on the Internet. The "publisher/platform" thing is pure horseshit spread by people who cry about "online censorship". Unfortunately that tactic seems to be working.

This is not correct. Section 230 does not restrict or amend a sites freedom of speech, nor does it make a distinction between a platform or publisher.

The effect of the law is that sites have less civil liability for user generated content, even if they make some decisions about what content is acceptable.

Those decisions can't get limited under the first amendment, and the content still has 3rd party speech protection.

So the government asking YouTube to remove a video would be a 1a violation, by controlling it's moderation decisions.

I am not firm with US law, and there is something I just do not understand - but would like to.

If YT is just some form of medium, not the "speaker"/provider and by that not allowed to remove content under its own freedom of speech, why is it then allowed (even forced) to block my content under copyright laws if it infringes someone's copyright.

Why not just route the DMCA request to me and I would need to react. Isn't YT treated as provider/publisher, when forced to react to DMCA requests on my behalf, if I infringe on some copyright?

My understanding is that the DMCA contains specific provisions requiring hosts to remove content upon request, and stripping them of their protections if they do not do so.

Youtube already has guidelines on the content you're not allowed to post (spam, nudity, violence, harassment, hate speech...) and that hasn't impacted their CDA or copyright-safe-harbor exceptions.