If you haven't already, this is probably a good time to switch to EdDSA keys. EdDSA signatures don't require RNG nor modular math unlike ECSDA signatures.
EdDSA signatures are specified to use deterministic nonce generation, so you're correct that they do not require randomness. But they certainly do require modular arithmetic in order to implement the elliptic curve operations!
Despite being fairly old, many systems still don't support eddsa. As an example, the reference browser/ca spec doesn't allow it: https://cabforum.org/working-groups/server/baseline-requirem.... Last I tried, even let's encrypt won't grant an ed25519 cert despite it being in tls 1.3 (2018?).
I don’t think RNG or modular math were really the culprits here. PuTTY’s k value generation is deterministic and the biasing was caused by a mismatch of integer sizes and the resulting leading zeros. The offending operation is named mod, so that’s related to modular arithmetic, but the modulo (521 bits) was bigger than the SHA512 output (512 bits) from deterministic k generation. I linked earlier to a post where I break this down at the source code level.