What is going to happen when it is totally infeasible to download the entire bitcoin transaction history?
The default answer to scaling is SPV (lightweight) clients, which allow very good security and control of funds without the entire blockchain. Full nodes will still exist but the cost of running one will not be prohibitive.
In practice many people will use some sort of bank service. If the funds are in full control of the bank it has some disadvantages, but yes - in this case of course the banks will settle intra- and inter-bank transfers with internal records and reciprocity agreements, not with the blockchain.
There are some schemes which allow making a payment without it being recorded as a transaction in the blockchain, and without having to trust a bank with your money. This will greatly reduce the total size of the blockchain.