Ecash has different scaling properties than normal ledger-based payment systems. Let me explain how a single mint can support peak volumes of millions of transactions per second.

Imagine a VPN service where users mint ecash and pay-per-use instead of pay-by-account (privacy!).

User pays 10k sats to a mint and acquires 10,000 x 1 sat tokens locked to the VPN provider's pubkey. This might take a second or two.

Once locked, the user could stream these tokens to the VPN provider for every kb of data they use.

Once the the provider checks the incoming token's signature, they can be sure that it can't be double-spent! They can remain offline (no round trip with the mint) and do this thousands of times per second (CPU-bound).

End of week, the VPN provider unlocks the ecash and redeems.

Now, this was one user with one provider. The mint didn't even notice the payments. In theory, the same mint could do this for 100s of providers with 1000s of users.

All it does is sign tokens (lock) and redeem them later (unlock). It's not involved in the payments themselves.

Note that this works because the burden of validation is shifted from the server (mint) to the user (VPN provider) at time of payment.

Also, since eventually all tokens need to be minted and redeemed at some point, the above only holds for peak volumes and not for average tx/s.