Thing is that, as you add more qubits, "operation fidelity" (as they call it) goes down exponentially, maybe even superexponentially (?), so it's very hard to keep it within an acceptable range on larger systems.
If you look at the theorem, it is definitely sub-exponential in the number of qubits needed to correct for error.
"Quantum mechanical states are extremely fragile and require near absolute isolation from the environment. Such conditions are hard to create and typically require temperatures near absolute zero and shielding from radiation. Thus, building quantum computers is expensive and difficult. The challenge increases dramatically with increasing size (number of qubits and the length of time they must be coherent) and thus only small to medium scale computers have been built so far.", from [1]. Also, I recommend reading out the whole article as it gives a nice, broad, overview of the challenges in place, it's not as easy as putting 1 qubit + 1 qubit together.
1: Quantum Computing: An Overview Across the System Stack, https://arxiv.org/pdf/1905.07240.pdf
Even in architectures with nearest neighbor gates, the (multiplicative) overhead stemming from error correction will only need to be logarithmic in the size of the computation. The constants may be unfavorable, but a log is still a log. See for example https://arxiv.org/abs/1208.0928 and https://arxiv.org/abs/1310.2984
The paper you linked to names challenges, but it only states those in respect to reaching practical scaling, which is a result of the constant that is associated with the growth rate, but not the growth rate itself.