How do I clean out Contacts without pressing Delete for each name in the list?

The easiest would clearly be to sign in with your Apple ID on the iCloud website, from a PC. In there you can see all your contacts - in there you can mark the contacts you want to delete, and finally, press the "Settings" icon in the bottom left corner, and select delete.

Sign in to iCloud here:

https://www.icloud.com/