Why skepticism must involve more than scientific testability

There has long been considerable debate about the matter of 'testability' of claims and the right approach by skeptics.  Here is an up-to-date example of this discussion:
http://www.skepticblog.org/2013/05/08/testable-claims-is-not-a-religious-exemption/

I'm definitely on the side of those who don't believe in any exemptions.  After reading the well-written article linked to above, you might wonder how I manage to disagree with it.  I disagree for two reasons.  The first is that supposedly untestable claims can have huge political consequences.  Someone can believe in God just because of faith, which puts their belief in the 'untestable' category for some skeptics, and that belief can lead them to all kinds of positions on moral questions.  So, these 'untestable' claims really do matter.  The second reason is that it's always possible to investigate claims even though the subject of the claim is supposedly beyond the reach of science.  We can take a look at the origin of the belief, such as the cultural and psychological influences, and we can look at what the claim actually means, and in doing so we can sometimes show that claim is nonsensical even though it is beyond physical investigation.  This was shown clearly by A.C. Grayling in his latest book 'The God Argument'.  The claim that the Christian God exists is nonsensical because that God has contradictory attributes; the label 'Christian God' doesn't refer to anything that could exist, no matter whether or not the existence could be scientifically verified.

There are more ways to be skeptical about existence claims than by using science, and we should not limit ourselves to debunking claims which are only testable physically.