How can you be sure that the public key on the cheque is funded to the face value though? When you're in the hypothetical desert/rainforest/mountainside situation, where internet access is too far away to do a blockchain query, or indeed to just do a straight-up transaction?
You can't, I pointed that out. You have to trust your partner. Just as you have to trust someone, who is issuing a cheque for you, no difference. But at least you can issue a cheque, and the partner can use it later without your participation.
This is just one of the possible options. We're comparing bitcoins to other means of exchange, not to some ideal magical currency
In the desert you got no phone banking either, and gold could be too heavy to carry.