...which drives up the shipping price, especially for international shipments...and is likely to be unclaimable anyway.

The following claims are considered to be non-payable by the USPS.
From the Domestic Mail Manual:

S010 Indemnity Claims
2.14 Nonpayable Claims

Indemnity is not paid for collect on delivery (COD), insured, or registered service or for Express Mail in these situations:

...
r. Negotiable items (defined as instruments that can be converted to cash without resort to forgery), currency, or bullion valued in total at more than $15 per shipment sent by Express Mail, except under 2.12c.
...


A simple technological means to prevent the eventual accident is always going to be better than paying for insurance to spread the cost of the accident among all the customers.

Common carriers sometimes already consider Bitcoins to be "currency", and they are definitely negotiable items.  There would be no recourse in collecting on an insurance claim.  It is clearly excluded.  FedEx recently returned a shipment of Bitcoins to the shipper because it said on the invoice it contained bitcoins and "we do not ship currency".