[cleanup] Remove Non-Pipelined Signer Logic

The change in !3372 (merged) has substantially reduced signing failures as expected (https://dashboards.ninerealms.com/#tss-health):

There have been a few spikes since the pipelined signer took effect early February, but these were due to unrelated P0s that prevented consensus on the outbound being signed (all known now fixed). The trend in the baseline error rate is clear from the chart, especially considering the outbound count has been higher during the period with the reduced error rate.