Module doesn't follow the Synapse logging context rules

[squahtx]

The code in this module doesn't appear to follow the Synapse logging context rules (https://matrix-org.github.io/synapse/latest/log_contexts.html) and drops logging contexts everywhere.

[squahtx]

This is making it difficult to understand logs in bug reports, eg. matrix-org/synapse#10397 (comment) matrix-org/synapse#12456 (comment)