I found someone asking the exact question I have, except the question and answers appear to be specific to SQL Server, even though that fact was ommited.
Here's the question Return a count of all foreign keys across all tables in database without a cursor
And what looks to be an elegant answer: https://stackoverflow.com/a/28729413/1402498
But I need to do this in Postgres. Is there an elegant way to accomplish the same thing in Postgres?