As it turns out, it is indeed possible to build an iPhone from scratch using a hodgepodge of parts, as Allen demonstrates in the video below.
He built a like-new 16GB iPhone 6s using components that were purchased in the cell phone parts markets of Huaqiangbei, China. The finished iPhone 6s is fully functional and comes complete with a working Touch ID Home button because the logic board and the Home button were purchased together.
Allen didn't save any money building an iPhone from the ground up -- on reddit, he says he spent "well over $1,000," but that ended up including extra parts, components that broke, or tools that were unnecessary. He thinks approximately $300 worth of parts actually went into the iPhone.
Because iPhone 7 parts were still difficult to find when he embarked on the project, Allen chose to build a previous-generation iPhone 6s. While most of the parts weren't too difficult to obtain, he says it was hard to get his hands on a logic board. He also had help from many of the vendors who sold the parts during the assembly process.
Allen outlines his experience building the iPhone in the video above, but additional details on sourcing the components and the assembly process can be found on his blog.