I recently had sent a Canon camera in for a repair. Somewhere in Canon the paperwork got messed up and they were going to charge me for a warranty repair. I don't have any idea what happened, but was able to get it straightened out contacting the right people.
I'm wondering if perhaps your repair paperwork got assigned to another camera that did have a physically cracked LCD and that may the cause of the confusion.
In another forum someone had sent a camera in for a repair and ended up getting one back that had a totally different serial number on the external sticker, as well as what was in the EXIF. He's now having a great deal of trouble trying get this straightened out since his purchase receipts and records don't match up to the camera that was returned. Mix ups happen. Make sure the serial number of the camera they are looking at matches yours.
I'm assuming you purchased the camera new. Is there a remote possibility that camera may have been gray market and thus Canon is looking for an excuse to not repair it. You would have had a US warranty card included if it's a US warranted model.
If possible, when you contact some one at Canon, try to get their email address and follow up the phone call with an email to that person with an overview of what was discussed. This isn't always doable, but worth a try.