Automatically redirect to your main client

Having had another bad onboarding experience during a discussion about not using MS Teams (I sent the "Copy room link" from the element web client), I thought about how to make this experience better.

Going a bit through the archives (#15 #197 #16 #74 https://github.com/flaki/matrix.to/commit/7eeadc4d21d569a86ea139b5301853ee694cdd53 #23 and others), I see that there are already quite some options, but neither of which makes it as easy as "click on the link"...

# Proposition

## Preparation of the web clients (e.g., element-web):

1. add a `default_client_storage` configuration option to element-web to point to a "matrix.to" server - by default it's off
a) there can be a per-user configuration option which overrides the global setting either way (on -> off; off -> on; override the URL)
b) for this to work, the inviter and the follower need to have the same `default_client_storage`
2. if the configuration is set, the element-web regularly opens a page from the `default_client_storage` server with an URL like:
`${default_client_storage}#mymatrix.mydomain.mytld` in the background. This is necessary, because the localStorage is private to a domain
3. This page from the `default_cliet_storage` stores the data after the `#` in the localStorage of the browser

## Sharing a room

1. click the three dots on the room and choose "Copy room link - default client" (or make this the default if `default_client_storage` is set)
2. the link points to the `default_client_storage` server with an additional tag, e.g., `?use_default_server`
3. if the page from `default_client_storage` detects a stored client in the localStorage of the browser, it shows a countdown (5 seconds) to redirect to the default client
a) if multiple default clients are stored, it could redirect to the most used after 5 seconds, but display the other ones

## Commets

- For mobile clients, matrix.to would need to recognize that it's running on mobile, and propose a rewritten `matrix://` scheme

# Problems

- need to open the link in the open tab, else the element web client complains that it's already open in another tab
- will not work with the desktop apps, as the `matrix://` protocol is not implemented yet, from what I gather
- this breaks my first idea of getting the web-client to store data in a privacy-preserving way (using an iFrame to store data in the localStorage): https://developers.google.com/privacy-sandbox/cookies/storage-partitioning

Do you think this would be an acceptable extension to matrix.io and element-web?

No files or tests are named. Start by reviewing the referenced issues and commit, then inspect the current matrix.to redirect flow and how element-web handles shared links, localStorage, and client selection. Done would require an agreed design and coordinated changes for default-client storage, redirects, and mobile or desktop behavior.

Written by the indexing model from the issue text.