I found a trick using a rule. I was, in fact, interested in adding myself to every card added to a certain list. So I added a rule:
"when a card is added into the list "My List" join the card"
(you can also do "add member @... to the card" if you want to add another member)
This will assign you to any new task on that list.
Now, the trick is that you can also reuse this rule for existing tasks: just move them to another list, then back to where they were. That makes 2 drag-and-drops per card (if you want to preserve the original card order, it may require a little dexterity though).
You can also press the keys <> or ,. to move the card under the cursor to the left/right quickly, but be careful, if you press twice without moving the cursor, it will move the same card. So you need to move the cursor a little before pressing <> or ,. again.
Alternatively, create the rule on a new list, and move all the cards to that new list. Then delete the old list, and rename the new list like the old list. That makes 1 drag-and-drop per card. Or 1 key press + small cursor motion if using <> or ,..
Note that there is a small lag before the rule applies, so you won't see the member icon appear immediately. But it will eventually.
If you do not want to keep adding members to new tasks, remember to delete the rule!