Investigate way to consolidate and display time spent on external calls in request logs

Context

This is a followup from !1088 (comment 1032683391).

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Quoting @igorwwwwwwwwwwwwwwwwwwww:

Tangentially related (though can be handled separately): It would be a good idea to have per-request counters and timers in our logs, so that we know which fraction of a request was spent waiting on redis or postgres.

In our rails logs, we have fields like redis_count, redis_duration_s, db_count, db_duration_s. I figured I'd mention it here, since that relies on instrumenting the same code paths.