This is what happens when websites prevent an open Console/DevTools side panel. They basically have a main loop running, inserting a debugger; statement in various places where they're annoying, and they do that at 30FPS / 32ms so that the DevTools become useless because there's no way to "ignore" debugger statements.
This button was added to specifically combat that, and even then it means you can't use breakpoints at all. IMO the missing link is a way to specifically ignore a single debugger call.
The debugger lets you edit the code in the VM, so if there's a debugger call you don't want you can simply delete it and continue and it'll not show up again until you refresh
Another way to do this, if you don't want to wait 5 seconds, is just to use the "pause script execution" shortcut, which is cmd-\. It'll freeze the page on whatever line of JS was currently executing, and then you can inspect the problematic element like normal.
Browser makers: please ignore 'debugger' statements by default with an opt-in in the dev tools. Almost all legitimate uses of the debugger statement can be replaced with a simple breakpoint if you're working on the code.
This is clever; after all, the only way to beat the recursive turtle stack of chrome debuggers debugging themselves is with the debugger statement.
sam.pl, of the infamous myspace Sammy worm, used debugging gotcha's to prevent visitors from de-mystifying his obfuscated html homepage.