Why do we send mining.set_difficulty in this case? Only to show the client we support this method?

Right.

And I can't think of any use case of the mining.set_difficulty subscription id. What was the original idea behind it?

Basically every subscription would get an id so you could unsubscribe from them.

In this case the default difficulty is 1 until the server sets a higher one, right?

Right. And the changes only take effect for notifys after it.

In the end wouldn't it make much more sense to set the difficulty per worker if we already support multiple workers per connection?

Yes, the stratum mining protocol is quite poorly designed.