I wrote the "...TV [being] scheduled to check for updates around this time..." not a cable-box. You can schedule some makes to check its website for updates. For 3 years, it could have either been a recurring failed, corrupt or incomplete update. I understand this sounds extreme and although you chose the word "patience," I would have chosen another. Resetting the TV to its factory defaults might help.
I don't feel it is an issue with the Cox signal...especially after 3 techs. The Cox signal would be strongest at the demarc. The primary feed into the house (living room) would be the closest to the demarc. I'm not sure where the mini-boxes get signals...either from a splitter at the demarc or in the living room...but I'll assume they're further (cable runs) from the demarc than the main cable-box in the living room. Again, just an assumption.
Peak network-activity during primetime could be an issue, but this congestion would affect Internet-streaming more than cable TV. I would assume CATV requires less bandwidth because it's probably more compressed, multiplexed, half-duplex and, at best, formatted as 1080i. OP reported streaming is okay.
The chronology of a regimented lifestyle could also be an issue. When do you normally turn on this TV? Is its coax and splitter near any electrical wiring or device emitting high EMI? It could just take some time...1 to 30 minutes...for the interference to finally disturb the signal.
Why would you guess 1 type of box would require a stronger signal than a mini-box? I'm sure the operational specs are documented somewhere, but have you ever compared? What are the operational specs of a "main box?"