Silver solder isn't pure silver - it's a mixture of metals with silver in it. The other metals mixed with it for what is called a eutectic mixture - this lowers the temperature of the mix to a point where it all melts.
And the reason silver is touted as better is that it has the lowest resistance among the elements. Whether the hype is true, I don't know.
One reason not to 'weld' parts together is that that would make it harder to remove the parts in case a repair is needed. Plus, like Czarivey pointed out, high temps would destroy everything.