A more apt comparison, instead of comparing the sudden destruction of the US or North America to the rest of Earh, would be the sudden destruction of Washington DC impact on the survival of the United States. Both Earth and Washington DC are the capital of their respective geopolitical entities. A sudden loss of either that completely wipes out the heads of state, congress, senate, joint chiefs of staff, military command HQs, etc... might have a destabilizing effect. Oh, sure, you might be able to beam up some governmental leaders, but that would require some advanced notice and time. What if the destruction was so complete and so fast that there wasn't time? Or that the rescue ships in orbit were also engulfed in the destruction and didn't have time to warp out (and we're ignoring ST Beyond's magical cross galaxy transporter).
The Federation isn't as united as the US. It seems to be more like the EU with more authority. It might be very likely that the sudden loss of Earth with all the heads of state for the Federation would result in the member nations contracting back into their own independent groups at least until the dust settles and new treaties could be established.
The Federation isn't as united as the US. It seems to be more like the EU with more authority. It might be very likely that the sudden loss of Earth with all the heads of state for the Federation would result in the member nations contracting back into their own independent groups at least until the dust settles and new treaties could be established.