[Feature] for embedding Click with custom stdout/stderr instances

Along with #2360, I also encountered another issue that there is no public API to override the stdout/stderr instances used by Click contextually and globally. The only way found is that manually replacing click.utils._default_text_stdout and click.utils._default_text_stderr private functions internally used by click.echo() and other console output handlers in Click.

Could we have a public API to achieve this?

import sys
from contextvars import ContextVar, copy_context
from typing import TextIO

import click
from prompt_toolkit.contrib.telnet.server import TelnetConnection

current_stdout: ContextVar[TextIO] = ContextVar("current_stdout")


def _get_current_stdout() -> TextIO:
    stdout = current_stdout.get(None)
    if stdout is None:
        return sys.stdout
    else:
        return stdout


def _get_current_stderr() -> TextIO:
    stdout = current_stdout.get(None)
    if stdout is None:
        return sys.stderr
    else:
        return stdout


@monitor_cli.command()
def help(ctx: click.Context) -> None:
    """Show the list of commands"""
    # You may think that we can just add `file` argument in this simple case,
    # but still there are other non-accessible internal usage inside Click
    # such as the default help option callback, etc.
    click.echo(monitor_cli.get_help(ctx))


...


async def start_server(...):
    click.utils._default_text_stdout = _get_current_stdout
    click.utils._default_text_stderr = _get_current_stderr
    ...


async def handle_telnet_connection(connection: TelnetConnection):
    ...
    # There may be multiple concurrent connections!
    current_stdout_token = current_stdout.set(connection.stdout)
    try:
        ctx = copy_context()
        ctx.run(monitor_cli.main, ...)
    finally:
        current_stdout.reset(current_stdout_token)

Start with src/click/utils.py, especially _default_text_stdout and _default_text_stderr, and trace how click.echo() and the default help option callback obtain output streams. Review the proposed contextual and global use cases, including concurrent connections through monitor_cli.main. Done means a documented public API replaces private-function overrides while supporting both scopes.