TLDR: Install Safari Technology Preview https://developer.apple.com/safari/download/
My console stopped working after upgrading to:
macOS: Big Sur 11.4 (20F71)
Safari: 14.1.1 (16611.2.7.1.4)
iPhone iOS: 14.5
Spent a few hours trying to fix this, resetting, rebooting, reinstalling, etc. until I found this comment suggesting to try Safari Technology Preview. My regular Safari still doesn't work, but Safari Technology Preview (Release 125 (Safari 14.2, WebKit 16612.1.15.1.12)) works like a charm.
Safari web inspector is only showing Sources, Console and Audit
Safari Technology Preview installs in parallel to the regular Safari, you can use it just for development. Find it here https://developer.apple.com/safari/download/