I've heard that centralized exchanges are more efficient, and that a centralized bitcoin system as a whole would allow for more transactions to be processed. Does that mean that within centralized bitcoin exchanges more transactions can be processed per second?
I have heard this point being made in discussion before, and I think you're missing a part of it:
The Bitcoin blockchain processes transactions in a trustless decentralized manner. This is due to the ledger of the payment system not being under control of any sole entity. This is an extremely inefficient and expensive process.
A centralized service can perform Bitcoin denominated payments much quicker, because they control their ledger: They update their own records to reduce their liability to one user and increase their liability to another. Nothing goes on the blockchain, it is all just in their own books. Since there is no other party directly involved, they can do millions of transactions between users on their own system without breaking in sweat. The disadvantage is obviously that the user has to trust the centralized service to pay back the liability at a future date.
A centralised exchange would still require a participating node in the Bitcoin network to process said transaction. I do not see how an exchange who processes Bitcoin transaction would achieve centralization that way. But suppose that an exchange is also a market-maker and said exchange maintains its own private, centralised ledger to keep track of who owns how many Bitcoins internally, then yes, said exchange would be able to process these internal swap transactions very efficiently as there is no need to relay this information to the Bitcoin network. The downside of this efficiency is the necessity of trust in said exchange. How can one be assured that said exchange will not screw you over one day, claiming that it was "hacked" and lost all the Bitcoins.