Ship logs to Posthog in stage

[ryanjjung]

We should do this in dev first, where we can play with some things without any real data or any changes to live servers.

[ryanjjung]

I posted this PR per our discussion in Infra Sync the other day to kick start a more solid plan around logging.

[ryanjjung]

This issue no longer represents our thoughts on this task, and it no longer represents the repo where the necessary changes will be made. I have created mailstrom issue #196 to track this work in a way that is more aligned with our recent discussions around logging, and I'm going to close this one.