Not the OP but I'm having the same issue. Happens on the windows app as well as on web.airdroid.com. I've tried the same steps as OP to fix it, no dice. The problem only seemed to start happening after a software update on my phone that seemed to add new functionality to my SMS on the phone itself (namely, seeing when messages are delivered and read, and seeing when the other person is typing). Not sure what the software version number is or when exactly it was released, but I'm thinking maybe the updates to the SMS borked something with AirDroid? Everything else about AirDroid seems to be working fine, including sending messages from my PC, it's just not showing received SMS messages.
That seems to be a rare issue... Can you please report this issue via Feedback with AirDroid on your device?
Steps: open AirDroid on your phone-->Me--> Settings-->About--> Feedback.http://forums.airdroid.com/viewtopic.php?f=4&t=28318
Please do tick the box of “Attach error logs” so that our developers can check the issue for you. It would help us to gather the troubleshoot with more details (there’s no personal data collected), we shall be able to prioritize your request.
Should you have any further questions, always feel free to let us know anytime!