WIP: Serve playground from correct route if nested APIrouters within one another #958

Job	Run time
Python 3.8 tests	17s
Python 3.8 tests	26s
Python 3.9 tests	17s
Python 3.9 tests	20s
Python 3.10 tests	21s
Python 3.10 tests	25s
Python 3.11 tests	19s
Python 3.11 tests	16s
lint / build (3.8)	12s
lint / build (3.8)	13s
lint / build (3.11)	23s
lint / build (3.11)	21s
pydantic-compatibility / Pydantic v1/v2 compatibility - Python 3.8	26s
pydantic-compatibility / Pydantic v1/v2 compatibility - Python 3.8	22s
pydantic-compatibility / Pydantic v1/v2 compatibility - Python 3.9	21s
pydantic-compatibility / Pydantic v1/v2 compatibility - Python 3.9	32s
pydantic-compatibility / Pydantic v1/v2 compatibility - Python 3.10	10m 13s
pydantic-compatibility / Pydantic v1/v2 compatibility - Python 3.10	10m 14s
pydantic-compatibility / Pydantic v1/v2 compatibility - Python 3.11	22s
pydantic-compatibility / Pydantic v1/v2 compatibility - Python 3.11	20s
	26m 40s

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