Yes, that makes sense. Most hard drive failures I have seen are between months and years. Sure there are some that will die upon installation or within hours, days or weeks, but it you have one of those drives (luck of the draw), you probably will not reveal any immediate impending failure by having it just spinning. You will need to have the head assembly(s) seeking, reading and writing to give it a test.
So let it start backing up and check the backup logs or file statistics if manually copying.
When I format a drive, given an option, I always go with the 'slow' format to uncover any bad sectors.
Steve