This is what worked for me...
My guess is that the issue was bandwidth on the local Wi-Fi, or possible signal strength, but more likely bandwidth.
I had tried reinstalling the app, resetting the APs, w/ home sharing, w/out home sharing, forgetting and reestablishing the remotes, etc....
I had been using a USB network adapter for connecting the iTunes machine to the Wi-Fi. The AP Extreme is on a different floor in the house. However, the adapter was wireless N and the signal strength was fine. Internet was fast but Remote was intermittent at best.
I plugged the PC into the time capsule directly via Ethernet, and now Remote connects to iTunes. Initially it takes a second. Then, it runs in the background on the phone and syncs quickly.
I thought upgrading from iPhone 4 to 5c would fix the issue with Remote, but it didn't.
Here is a screen capture from the APU on miPhone.
After all is said and done, the average user probably wouldn't have made this work. Prior to iTunes 11, we never had a problem.