Unfortunately, there appears to be an issue with the Mac and AirPort Utility as far as its ability to discover the Time Capsule on the network. No doubt this same issue is what was causing the "unknown errors" when you tried to use the Mac to configure the Time Capsule.
So, the initial impression might have been that you were having a Time Capsule issue, when in fact the real problem is that you have a Mac issue. The fact that you could configure the Time Capsule with an iPad or iPhone pretty much confirms this.
Mavericks (10.9.x) and Yosemite 10.10.x) are both known to have these kind of bugs with "lost" Time Capsules and other AirPorts as well.
We can try to check a few settings, but cannot offer any guarantees that we'll be able to find a way to work around the bugs.
On the Mac.....open System Preferences (gear icon on the dock), and then open Sharing. Check to see that the name of your Mac appears correctly here, and that a (2), (3), (4), etc does not appear after the name. If it does, edit to remove the (2), (3), (4) so the name of the Mac displays correctly.
Open System Preferences again if it is not already open and open Network
Click on WiFi on the left panel to highlight it, then click Advanced at the lower right of the window
Click the TCP/IP tab
Make sure that the setting for Configure IPv6 is set to Link Local Only. It should be, but strange things happen with Mavericks and Yosemite.
Click OK, the click Apply in the next window
Now click on Ethernet to highlight it, as you did with WiFi in the steps above, then click Advanced
Click the TCP/IP tab
Make sure that the setting for Configure IPv6 is set to Link Local Only
Click OK, then click Apply in the next window
Close the windows and restart the Mac
Pull the power cord from the back of the Time Capsule, wait a few seconds, then plug it back in and let the Time Capsule restart.
See if that helps to get Time Machine going.