PR at: https://github.com/python/cpython/pull/7874.
I re-ran benchmarks since shutil code changed after #33695. Linux went from +13.5% to 8.8% and Windows went from +17% to 20.7%.
In the PR I explicitly avoided using a context manager around os.scandir() for now so that patch it's easier to review (will add it before pushing).

Linux (+8.8%)
=============

without patch:
    $ ./python  bench-copytree.py 
    Priming the system's cache...
    7956 files and dirs, repeat 1/3... min = 0.604s
    7956 files and dirs, repeat 2/3... min = 0.603s
    7956 files and dirs, repeat 3/3... min = 0.601s

with patch:
    $ ./python  bench-copytree.py 
    Priming the system's cache...
    7956 files and dirs, repeat 1/3... min = 0.557s
    7956 files and dirs, repeat 2/3... min = 0.548s
    7956 files and dirs, repeat 3/3... min = 0.548s
    best result = 0.548s

Windows (+20.7%)
================

without patch:
    C:\Users\user\Desktop>cpython\PCbuild\win32\python.exe cpython\bench-copytree.py
    Priming the system's cache...
    7956 files and dirs, repeat 1/3... min = 8.275s
    7956 files and dirs, repeat 2/3... min = 8.018s
    7956 files and dirs, repeat 3/3... min = 7.978s
    best result = 7.978s

With patch: 
    C:\Users\user\Desktop>cpython\PCbuild\win32\python.exe cpython\bench-copytree.py
    Priming the system's cache...
    7956 files and dirs, repeat 1/3... min = 6.609s
    7956 files and dirs, repeat 2/3... min = 6.609s
    7956 files and dirs, repeat 3/3... min = 6.609s
    best result = 6.609s