Documentation is misleading for add_url_rule when using AsyncApp

[fcollonval]

Description

The typing expected by starlette route callback is:

route: typing.Callable[[Request], typing.Awaitable[Response] | Response],

https://github.com/encode/starlette/blob/6f863b0d3b8e8f18d5df9e8cd2514f7085b874e1/starlette/applications.py#L161

So the example shown in the doc (that works for flask), does not work for AsyncApp

def healthz():
    return 200

app.add_url_rule("/healthz", "healthz", healthz)

Expected behaviour

add_url_rule works the same way for Flask and AsyncApp and connexion context can be used.

Actual behaviour

Currently to make it work

from starlette.request import Request
from starlette.responses import JSONResponse

def healthz(request: Request):
    return JSONResponse({}, 200)

Steps to reproduce

Add a custom rule using app.add_url_route

Additional info:

• Python 3.10.13
• Connexion: 3.0.6
• starlette: 0.37.2