Big talk, but when the issue of Ukraine in NATO comes up seriously amongst the majority of NATO, I doubt that Slovakia has the influence or endurance to resist the pressure of the entirety of the rest of the alliance. There's a reason that the only real* holdout on Finland and Sweden was Turkiye, and Turkiye has the advantage of being incredibly important to NATO. Unlike, say, a landlocked country in the heart of Europe surrounded by NATO members and NATO allies.
Adding a new member to NATO requires unanimous approval from the existing members, so strictly spaking Slovakia does have the influence. That said, I expect that in the same way thar the other countries were able to cajole and persuade Turkiye with things like suspending arms sales, and Slovakia would surely be easier to influence
The leader of a country that was over-run by Russia and thousands murdered and starved in the Soviet years wants to risk losing their NATO status by preventing Ukraine enjoying the same status. I'll bet that'll go over like a fart in church with the people in (Czecho-)Slovakia that remember that history.
Kick Slovakia, Hungary and Belarus out of the EU and Nato. Lets them become Moscow's slave countries again and see how much they enjoy getting what they seemed to have wanted.