Since you apparently have extremely strong and direct broadcasts, with reception not requiring any antenna whatsoever on channel 11, and physical line of sight to the towers from your property, you should not encounter dropped frames on this channel unless you have some very strong multipath from reflective surfaces between you and the transmitter tower. This should be very apparent if you connect your log periodic directly to your TV input without a splitter and without Tablo.
Even if signal strength and signal quality are creating problems for you, this in no way should cause your Tablo to disappear from your network and require rebooting. All logic tells me that your Tablo is defective if it worked properly for about a month (as you say) and then began disappearing from the network.
To the extent possible based on your network topology, I would connect the Tablo as close to the router as possible using a very good quality ethernet cable, avoiding intervening IP hardware such as switches/hubs etc.
I use twoTablos here, each with an antenna pointed towards different population centers/towers. Depending upon the length of the RJ-45 ethernet cable connecting each Table to my router, and the quality of the ethernet cable (I am now using cat 6), I can provoke problems in stuttering, lease renewal issues, dropouts, etc. One of my Tablos (a 4 tuner) is definitely more finicky than my other unit, a 2 tuner unit. The dependency on the length and quality of the ethernet cable could very possibly be attributable to a marginal ethernet transceiver chip in the Tablo.
My advice, for whatever it is worth, would be to run short, reliable RJ-45 cable to your router directly from your Tablo. If the disappearing Tablo persists, I would request a replacement unit.
In several years of use with a relatively complex environment here, I have never seen any reason to associate weak signals/poor signal quality with the disappearance of the Tablo from the network. these are two independent and unrelated issues as far as I'm concerned.
I would add that hard disk drive compatibility has in the past been a real issue for me, and that using Western Digital passports entirely solved my earlier problem in that regard.
One other cautionary comment:
Adding an amplifier at your antenna output can create problems in the presence of very strong signals (channel 11) . Since you are apparently a ham radio operator (I've been licensed since 1957), you'll appreciate that Intermod and 3rd order intercept performance of these amplifiers (particularly the low cost models) can wreak havoc with ATSC Digital television signaling.