If the paper is to be believed you are wrong in two different ways: 1) If the paper is right, they are claiming they need 300ish physical qubits that can susta

If the paper is to be believed you are wrong in two different ways: 1) If the paper is right, they are claiming they need 300ish physical qubits that can sustain about 1000 gates before decohering. No need for scalable error correction. 2) Independently of the veracity of the paper, if you actually need logical error corrected (and fault tolerant) qubits, you need error correcting codes with much more severe overhead than the 5-qubit code. The 5-qubit code is a pedagogical example, not something that would actually work under realistic conditions. And even the 5-qubit code needs quite a few extra ancillary qubits for fault tolerance (which is more expensive than simple error correction).