It really seems to be an iPhone 6 software problem based on all these reports. It is true that various fixes (resetting network settings, etc.) work, but only temporarily--in my case, for up to 12 hours. But then the same problems occur all over again. As with everyone else, all the other Apple devices in my household work just fine using iOS8 (iPads and iPhones), and our network is up to date and running WPA. (Besides, this issue also happens when I am on other wifi networks, such as the one at my office.) As mentioned, I've also done everything Apple suggested, including wiping the phone multiple times, deleting and reinstalling the software, and exchanging the phone. I also tried various ways of building the phone back slowly: leaving it at factory settings except for signing onto a wifi network, signing into the Cloud but not restoring any backup, Cloud-only restore, and iTunes's restore. I noticed that there is no issue if I keep the phone on my wifi network with factory-only settings and do NOT sign into my iCloud account. However, once I sign into the Cloud, even if I do not restore my settings, the phone starts dropping wifi and reports to me that it is "unable to scan for networks" when I try to add my network manually. After some time, which is usually about 5-20 minutes, it spontaneously recovers the wifi signal.
Maybe it would help if we could figure out some common denominator among people who have this issue. For example, I have thousands of contacts saved and thousands of calendar events, each of which automatically loads onto my phone once I sign into my iCloud account (whether or not I restore from a backup).