Excellent analysis. Agreed that valuing Square as a credit card processor makes for a very frothy, perhaps unjustifiable, valuation.
My belief, however, is that some investors are betting on Square's potential as much more than a credit card processor -- really, more of a data and payments platform. If Square can own the marketshare of payment transactions happening offline, then other monetization opportunities appear and the credit card processing is more of a loss leader or infrastructure investment (e.g., Amazon investing in warehouses and supply chain management).
Some of the other opportunities include: coupons (they know exactly what you buy and can tailor deals); analytics (e.g., household good purchases dropped by 25% in San Francisco last month); recommendation engine (I ate at these restaurants last month, this week Square recommends these); and a replacement network for credit cards, though this seems unlikely in the near term because of the difficulty factor and the fact some big banks are investors in Square.
While there are assuredly other ideas, and while some of these may never yield profits, the macro point is that investors are betting that Square can monetize payments data and payment identity in new ways -- if it can become the new funnel through which everyone pays businesses in the offline world.