Does it make sense to block RSA keys under 1024 bits?
Those findings were based on a broken PRNG. A broken PRNG affects all keys generated on such a device, no matter the size or the algorithm.

Common primes were how the problem was detected, but the problem itself is unrelated to primes or RSA. If RSA keys of a specific size where affected, that's only incidental because that's what the broken devices generated. 

Blocking keys smaller than 1024 bits can be a good idea, because their security margin against factoring attacks has grown pretty thin, even when they're correctly generated. But that threat isn't related to the paper you link.