It's not so much about power, but about the ad-hoc nature of attaching a debugger. If you're not there to catch and treat the error as it happens, a debugger is not useful in the slightest: by the time you can attach it, the error, or the context where it happened, are long gone. Not to mention, even if you can attach a debugger, it's most often not acceptable to pause the execution of the entire process for you to debug the error.

Especially since a lot of the time an exception being raised is not the actual bug: the bug happened many functions before. By logging key aspects of the state of the program, even in non-error cases, when an error happens, you have a much better chance of piecing together how you got to the error state in the first place.