[3.13] gh-128479: fix asyncio staggered race leaking tasks, and logging unhandled exception.append exception (GH-128475) (#129227)

gh-128479: fix asyncio staggered race leaking tasks, and logging unhandled exception.append exception (GH-128475)
(cherry picked from commit ec91e1c2762412f1408b0dfb5d281873b852affe)

Co-authored-by: Thomas Grainger <tagrain@gmail.com>
Co-authored-by: Peter Bierma <zintensitydev@gmail.com>