So far I can only find information as to why you want to increment the versionCode (for Play Store purposes), but I cannot find any information on any pitfalls that might be a result of not incrementing it.
Currently it is more advantageous for me to keep the versionCode at 1 so that I can easily downgrade the application without uninstalling the newer version first if the need ever arises. Is anyone aware of any issues not incrementing the versionCode might cause outside of the Play Store?
There are no "issues", except for the fact that you will never know what actual version of the software is installed on any given device if all the versions of the software have the same versionCode. This is just standard, recommended, best practice in software development: You should always be able to tell exactly what code is running on any given device at any given time. If that isn't necessary for you, and you see advantages in leaving the versionCode unmodified, then go right ahead.