Bitcoin doesn't need "everyone processing everyone else's transactions", only full nodes need to check transactions for validity. Bitcoin's whitepaper proposes SPV wallets, which are meant for end users and require minimum disk space and network bandwidth, and no need to run a full node.

I recommend this technical article on how Bitcoin (in its Cash fork) can scale to worldwide adoption (50 tx per day for each of 10 billion humans) even on today's hardware: http://blog.vermorel.com/journal/2017/12/17/terabyte-blocks-...