The larger issue with identity theft than the inconvenience of proving that the transactions aren’t yours, is that it can lead to the victim being charged criminally and arrested.

I knew someone who had her identity stolen, she took all of the proper steps - going to the places to prove it wasn’t her, filed police reports, etc. and she still got arrested for various fraudulent activities like check kiting and theft by conversion. She ended up in jail for a weekend while relatives brought various forms of proof that there was already an ongoing investigation. She had a warrant out for her in another county that she didn’t know about.