You are absolutely right.. when apple changed significant features in Lion and Mountain Lion .. they introduced a bunch of issues into networking which are bugs to this day..unacknowledged.. and without much that will fix them.. other than your reboot strategy.. which is like.. well that other OS.. where continual reboots are considered normal.
Old TC may also just suffer from age.. we continually tell people they have a lifespan.. 3years is the reliable span.. after that expect anything.. sudden death.. slow decline.. continual hassles and problems. If the backups are important to you.. then it is telling you in a very clear way.. replace me.. I am no longer happily providing services promised.
Now to specifics.. we don't know what the problem is.. if we knew we could give some work around.
The problem is odd in that it doesn't affect everyone.. if it starts .. people find it starts several months in to ownership.. it is never cured.. or even curable.. something like DNS or ARP table is being lost.. but it is not either of those things.
Is there a diagnostic routine I can run or any error detection I can look for?
Nothing much shows up anywhere. Including the TC log or local logs. It simply has lost the TC disk but not internet.
Some things do help.
I don't know if you have pondini's resources.. TM is a complex piece of software.. it has multiple issues.
Look at A1 .. load the widget.
A5 .. verify the backup.
A4 .. reset TM.
All of those things can help.. look at some of the typical problems.. C12 and C17 being the most common since Lion.. and exacerbated in Mountain Lion.. not fixed.
1. Keep names short, no spaces and pure alphanumeric. See C9 http://pondini.org/TM/Troubleshooting.html
2. Do not use 7.6.3 firmware... go back to 7.6.1 or even better 7.5.2 on an old one.
Firmware downgrade is easy in new utility. Hold down the option key whilst selecting version.
3. I recommend a very short dhcp lease time.. 20min for example. In whatever device is the router .. that can help keep things up to date.. and use dhcp reservation again to see if the address info helps to keep things updated.
4. Do turn on ipv6 local link.. which you need to anyway with 6.3 utility.
5. If you are using a desktop.. an external drive is faster.. more reliable and easier than TC for Time Machine backups..
6. Consider using CCC or another backup software.. if you want to continue to use network drive.
7. Although using 6.3 airport utility has some advantages.. it is also devoid of a lot of tools that are in v5.6 airport utility.. well worth loading into at least one computer on the network.. if you only have Mountain Lion now here is one method.
How to load 5.6 into ML.
1. Download 5.6 for Lion.
Click to open the dmg but do not attempt to install the pkg.. it won't work anyway.
Leave the package open on the desktop so you can see the file. AirportUtility56.pkg
2. Download and install unpkg.
Run unpkg on the desktop.. If your Mac refuses to run the software, because it wasn’t downloaded from the Apple store, go to security in preferences and allow other software to work.. this is limitation of trade methinks. You can set back later if you like.
Now drag the AirPortUtility56.pkg file over to unpkg.. and it will create a new directory of the same name on the desktop.. in finder, open the new directory, drill down.. applications, utilities .. there lo and behold is Airport utility 5.6 .. drag it to your main utilities directory or just run it from current location.
You cannot uninstall version 6 (now 6.3 if you updated) so don't try.. and you cannot or should not run them both at the same time.. although I have had no problems when doing so.
For screen shots see this post.