Well, the Klingon and Romulan Empires are Empires. They conquer and subjugate other sentient beings. If the centers of their Empires collapsed it would be the signal for their conquered territories to rebel and fight for their independence again while the military would be too busy dealing with whatever destroyed the Homeworld to keep things under control. With the Feds, it's a multi-species cooperative and not an Empire. So losing one of the Founding Worlds (there are 4 others) wouldn't be SUCH a blow by comparison.
The Federation I think could survive loosing 2-3 of its core worlds. (Earth, Vulcan, Andoria (if that is the right name) Teller prime, and some of the bigger newer members, like Betazed and Trill someworlds. ) However even a serious blow to Romulan or Klingon homeworlds would cause a major civil war, not only between the species uprising, but also from the remaining main species fraction. Also you would have species like the Breen raid in such cases.
Earth seems to be pretty darn important to the Federation, beyond being even a founding member - even the alien president seemed rather invested in the paradise of Earth. I don't think it's vitally important to the Federation politically, though, asides from being the capital, and thus being where all the important politicians are. However, Earth does seem to be the keystone of Starfleet. That would make it an important place for any attack, and why Starfleet is so keen to defend it (as opposed to, say, Betazed). As Starfleet is the defensive force of the Federation, Earth is important to the Federation. Therefore, Layton would want to defend it, even to the point of martial law (and of course, he's the one most qualified to lead it, since he's the only one taking the threat seriously...).