I've designed TL074s into various bits of home made kit many years ago, and despite being sold as low noise devices (and they are supposed to be a lower noise version of the TL064) I found then noisy.

This has nothing to do with balanced vs unbalanced. Changing the supply voltage and decoupling are unlikely to have any effect.

Adding a preamp before the box will help, but getting a better DI box in the first place would be a lot less messy and maybe cheaper in the long run.

The NE5534 is a great low-noise op-amp (and has been for nearly 40 years), but will take more current, and it won't be a drop-in replacement as it takes a lot of input bias current, while the TL064/74 family are FET input taking almost no bias current at all.

Would a passive DI do the trick? If their mixer has good low-noise inputs and they don't have instruments with high impedance outputs (e.g. some electric guitars/basses) they might be good enough.