What is it about piezo pick-ups on strings (violin, cello etc) that makes them sound so horrible?

It's not a property of the pickup itself being "bad". Rather, we do not typically listen to violins with our ears crammed against the bridge. We listen with the instrument in front of us and resonating from the strings and body through physical air. You're just not often going to get the natural, "true" sound of a string instrument from a pickup. In a studio setting I always lean on an overhead mic first and a pickup second if available. The pickup blend with the mic'd sound can help with some presence to cut through a mix but it isn't particularly necessary. It seems like you're primarily talking about a live setting in which case I would abandon hope of an "acoustic" sound without additional miking. Think of it as an acoustic guitar with a pickup, that doesn't sound natural either, they also often sound harsh and bright, but it's somewhat of a necessary evil when you're dealing with potential feedback.

/r/audioengineering Thread