Was the Haptic engine included as part of the iPhone's Home Button ever made available for use by developers?
Yes, Apple made that API available in iOS 10 (released in 2016). The main source of documentation is contained within the documentation for the UIFeedbackGenerator class:

https://developer.apple.com/documentation/uikit/uifeedbackgenerator

Many apps make use of it. For example the Halide app uses it to tell you, when you have the phone perfectly in "sync" with the grid. The Reddit app does it when you vote on a comment. The NBA 2k14 app uses it to indicate when the ball is touched. And many more apps/games exist that use haptic feedback.