It is fine only for junior engineers. For people on the higher end of the ladder, you have to let them know what type of roles you are recruiting for, and see if they are willing to make a move. Some will make the move if the job is an upgrade. This recruiter doesn't get it.
The recruiter is just bad at his job and makes amazon look bad. He/she is bad because amazon probably just requires hard quotas, and doesn't care about relationship building.
Every last job I have had, it was a recruiter sending me a message, then maybe months later, when i was ready we went through the process.
This recruiter is destroying the long term pipeline of the company for short a term quota. since this is amazon, I am not surprised of it.