Most of that is 100% correct James, I work for a very large communications company that has the initials of VZ (not VZW however). I main job is working at cell sites, installing and maintaining FO (fiber optic) cable and mux's (digital multiplexers)-in other words the "ILEC" in your description. We provide high speed "landlines" or "back-haul" circuits for various cell carriers like Verizon Wireless, ATT, Sprint, T-Mobile, Cricket, Nextel (now pretty much all those site are off-air and abandoned now). We provide T1's (not used much anymore), TLS (ATT and some Sprint) and GIGE (VZW) services.
The one thing I will disagree with is that at least in this area, MOST cell towers are owned by a third party that just build and lease space on towers, around here there is American Tower Corporation (called ATC) and Crown Castle International, they own probably 80% of the sites/towers between them. They pay the property owner, secure right-of-ways and easements, negotiate 24/7 access, and lease ground and tower space to the carrier. Some carriers do pay building owners, like churches and farmers directly for putting antennas in steeples or on silo's. On most towers, there are multiple carries.
Ductape: Sprint has very few "technicians" that actually work for them, most are contractors that are probably not getting paid enough to troubleshoot crazy problems like this. I suspect one of two things are happening:
1) Your signal is "to strong", meaning that the receiver is getting overpowered. Ever mess with CB's in the old days?
2) You are suffering some the "umbrella" effect, meaning that the signal is going over your house and missing you completely-I have seen/experienced this myself while at cell sites. But looking at the picture, and base on the fact that you have full signal strength I doubt this is the cause.