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Maybe you don’t want to touch the implementation of the "older high-level API" for fear of subtly breaking something, but for clarification, and perhaps documentation, would the old functions now be equivalent to this?
def call(***):
 # Verify PIPE not in (stdout, stderr) if needed
 return run(***).returncode
def check_call(***):
 # Verify PIPE not in (stdout, stderr) if needed
 run(***, check=True)
def check_output(***):
 # Verify stderr != PIPE if needed
 return run(***, check=True, stdout=PIPE)
If they are largely equivalent, perhaps simplify the documentation of them in terms of run(), and move them closer to the run() documentation.
Is it worth making the CalledProcessError exception a subclass of CompletedProcess? They seem to be basically storing the same information.