No, it is not the slippery slope fallacy. If you create an instrumemt that obligates fact checking, you have to give someone authority to define what are facts and what arent. And as this is obligatory by law, these fact checkers are subject to supervision or are directly part of the government.

So now the government gets to decide what are facts and what are not. Which can easily be abused. Especially as disinformation through so called fact checkers can move as fast as any other disinformarion.

So at the very least you need to create a sanction regime, e.g. criminal punishment for the abuse of the fact checking, as well as a right for people to have the fact checking checked and challenged, if they think it spreads lies against them. This way you can have it analysed by courts, as the most neutral authority in a state of law.

I dont get how people in Europe, where i live by the way, especially with the experience of Mussolini, Hitler and Franco fascism, as well as all the Warsaw pact authoritarianism, GDR surveillance, red scare policies in the Western countries during cold war, etc. are just treating this so lightly.

Authoritarian regimes based on lies and forbidding the truth are not some abstract. They are both an extensive reality of the recent past as well as looking at Orban, Melloni, Wilders, Merz and many others they are reemerging right now.