It’s certainly expensive, and the cost grows exponentially with the number of bits, but I don’t think say a 4-bit grind is beyond the power of small devices, which would suffice to leak a secret with 64 signatures.
Well the device could have been constructed maliciously in the first place?
Using FEC codes that’s not necessary. Using that, one can take the secret-to-be-leaked, expand with a several GiB “checksum”, and then in every signature leak a few deterministically-selected (e.g. based on the message and device key that’s only known to the hacker and/or malicious manufacturer) bits of that checksum. Given the size of the checksum, it’s unlikely that any two signatures collide in the position, and as soon as enough bits of the checksum are leaked (regardless of where they are), the attacker can reconstruct the original secret.
(Note that this several GiB checksum isn’t actually ever materialized; arbitrary bits of it can be computed on-the-fly).
The device could choose to only behave maliciously in certain situations, such as when interacting with large amounts.
If there are scenarios where interactive anti-exfil is just not possible, then none of the above are reasons not to proceed with a scheme like this, as it’s probably the best possible. But if people are seriously trying to address the exfiltration scenario, then I wouldn’t just dismiss the possibility that exfiltration is still possible through grinding.