The earlier you remove usages of external deprecated functionality, the more versions of your product will work with your dependency once the deprecated features are fully deleted or hidden in the dependency.
For example, api-common-java 1.6.0 refers to
in Guava, which was deprecated in 19.0 and removed in 26.0. In
api-common-java 1.7, the code was changed to instead call
Futures.transform(ListenableFuture, Function, Executor),
which is not deprecated. The oldest version of api-common-java that
can be used with Guava 26.0 or later is 1.7.0, because 1.6.0 and earlier
refer to a deleted method. If api-common-java had moved to the new
method even earlier (for example, if this had been done in 1.0.0), then
more versions of api-common-java would be compatible with Guava 26.0 and
later (1.0.0+ instead of 1.7.0+), making it easier to find compatible
combinations of versions.
This best practice refers to code that uses deprecated functionality from another library or the JDK, not to the deprecated functionality itself. It does not apply to usage of deprecated features within a library.