The transaction limit and public-ness of transactions are implementation details of certain blockchains. There are blockchains with much better TPS, and blockchains where transactions are not public.
Yes - you're missing "mathemagic" :) And a variety of independent engineering solutions which are largely independent of the mathemagic(think sharding or something like the lightning network).
Check out ZCash for an example of non-public, decentralized block chain.
