World wide safety net: Open borders isn't the same thing as everyone having the same safety net in whichever destination they travel to. Though related, they are distinct issues that can exist independently.
That said a world wide safety net would be a good thing to advocate for on moral grounds, no?
The EU has open borders for all EU citizens, and for the most part it's faring ok. There has not been a Europe-wide social safety net crisis as a result of this freedom, and crucially for this argument: there isn't an EU wide social safety net. It has freedom of movement without a universal safety net.
The USA's states are effectively countries. All USA citizens have open borders to travel between states in the USA. For the most part, it's faring ok and there has not been a USA-wide social safety net crisis caused by this freedom either.
Iceland and cheese for everyone: That sounds like what's called the slippery slope fallacy. https://en.wikipedia.org/wiki/Slippery_slope