A pretty small Redis server can handle 10k clients and saturate a 1Gbps NIC. You'd need a pretty heavy duty Postgres database and definitely need a connection pooler to come anywhere close.
I agree that redis can handle some query volumes and client counts that postgres can't.
But FWIW I can easily saturate a 10GBit ethernet link with primary key-lookup read-only queries, without the results being ridiculously wide or anything.
Because it didn't need any setup, I just used:
SELECT * FROM pg_class WHERE oid = 'pg_class'::regclass;
I don't immediately have access to a faster network, connecting via tcp to localhost, and using some moderate pipelining (common in the redis world afaik), I get up to 19GB/s on my workstation.
Sorry, I should have used something more standard - but it was what I had ready...
It just selects every column from a single table, pg_class. Which is where postgres stores information about relations that exist in the current database.