I know how GPS works. Its a constellation of 31 active sattelites in a medium-high orbit, your device passively listens to their signals, and 3 signals are enough to trinagulate your position.
Unfortunately its owned by the US military, so in the past they made it run less precisely in active war zones and near important military bases by introducing small errors, so in those cases it only worked nicely for US military equipment, because only that has the technology to fix the small errors.
Officially this is not being done anymore, it would also be pointless now because the civilian european galileo-system works everywhere. So it should even work properly near a military base. For the signal, woods are transparent. It really only fails if there are large amounts of rock, soil, water, or metal above you.
Maybe the GPS-antenna in your phone is broken and the phone compensates by doing pseudo-GPS with cell tower triangulation. Your internet will work fine with just one cell tower in range, but then the triangulation fails.
I googled it, apparently the GPS antenna in the iphone 5 can break, and it can be replaced. Apparently the phone can fall down hard enough to damage the connection to the GPS antenna but not hard enough to damage the display or anything else, then you have a phone where everything except the GPS works just fine.
Replacing it seems really really difficult, i found a video showing how it is done. Apparently you need professional level soldering skills and equipment, so the most useful aspect of the video may just be that it shows what the iphone 5 GPS antenna looks like.
https://www.youtube.com/watch?v=Q3gw1KDWuPw
There should be ways to determine if the GPS antenna is broken using just the firmware of the phone, or maybe a diagnostic tool, otherwise i do not see how apples service or a phone repair shop can tell if the antenna needs to be replaced.
Anyway, the easiest solution is still to borrow a phone from a good friend or family member that also has pokemon go installed, and use that to take the arena. Bringing the phone to an apple store, just for diagnostics to see if the GPS antenna is broken, may also be a good idea. Replacement will cost money, but if you just want it diagnosed because you suspect it is broken, i think they do that for free.
EDIT: Further research confirms my suspicion. An iphone 5 with a broken GPS antenna will compensate using cell tower triangulation, and it will not notify you, so you only notice that it is broken in remote locations.
That makes it extremely likely that your GPS antenna is broken, the problem you describe is a perfect match with what you should expect from an iphone 5 with a broken GPS antenna.