Can anyone shed any light on the reasons for this being considered private?
The org.infinispan module is private simply because there are no API/source or binary compatibility guarantees across releases. Additionally, you cannot easily upgrade Infinispan independently from WildFly itself.
Are we going to have issues in the future if we continue to use the built-in Infinispan configuration inside of Wildfly?
When upgrading to a new version of WildFly, it is possible that your application might not compile without source code changes. However this is true whether you bundle Infinispan within your application or use the version contained in WildFly.
Generally, using a server managed Infinispan cache is easier to configure/use as it can leverage other server managed resources for its dependencies, e.g. JGroups channels, transaction manager, DataSources etc.