Replaces #6457
In preparation for #6449

This PR fixes a couple of console output issues and logger setup issues, and also splits the console output into two streams. The separation of console output streams is necessary in order for the download progress to be fixed later on. Progress output should never be written to a file, but since we want some console output to be written to log files as well, especially error messages, we need to use two streams.

Currently on master, the ConsoleOutput always inherits the stream of Streamlink's root logger. This is problematic because --logfile=... sets the logger stream to a file, which means all ConsoleOutput including user prompts are written to the log file instead of being written to a TTY where the user has to write their input.

Also, --quiet currently does only affect the log level, as it's treated as an alias of --loglevel=none, which means the inherited output stream is still stdout, so all console messages and error messages are printed regardless. --quiet should mean no output at all.

Another problem is that if there's no TTY output for user prompts, those prompts currently don't fail. If the user doesn't have a chance to know what's prompted, then the prompt must fail. And failure is currently also not handled correctly and None is returned instead, which is wrong.

I've decided to use one single PR branch for these changes, because they are all a bit intertwined. Splitting this up would make this unnecessarily complicated for me.

So, a quick overview of the individual commits:

1. Fix the method name of the ConsoleOutput password prompt according to Streamlink's public interface of the UserInputRequester
2. Fix the prompts, raise OSError on failure, and most importantly, check for valid input and output TTY streams (must exist and must not be file streams).
3. Update the error handling of the one function in Streamlink's CLI that doesn't use the Streamlink session's UserInputRequester and that instead calls the ask() method of the ConsoleOutput directly.
4. Refactor ConsoleOutput and use two separate streams, one for console output and one for file output. Both are optional, as stdout/stderr can be missing, or there's no log file stream. Also ensure that the file output is not a TTY, so that --logfile=/dev/tty doesn't duplicate console messages (as msg() and msg_json() write to both streams).
5. Refactor the setup of ConsoleOutput and Streamlink's root logger and actually use two separate streams. Make the root logger inherit the primary stream of the ConsoleOutput (not the other way around). If there's a log file, attach the file stream to the ConsoleOutput instance.
6. Check if the output stream for Streamlink's root logger actually exists. Previously, the logging module of Python's stdlib switched to stderr if the passed stream was None, and we also had to check if stderr existed and use an additional fallback to devnull as a FileHandler in that case. Simply don't attach any log handlers to the root logger if there's the passed stream is None.
7. Make ConsoleOutput fall back from stdout to stderr if stdout doesn't exist, to align with the previous fallback mechanic of the root logger that was removed in the previous commit. Since the primary ConsoleOutput stream is inherited by the logger now, the old behavior is restored. But now, both console and log output have a fallback.
8. Make --quiet suppress all text output, not just log output.

Considering that there are a lot of changes, this needs a bit of additional testing before I'm going to merge this. As always, I've added tests with full coverage for the changed code parts, but this doesn't cover all potential combinations of CLI arguments and environments.