You do have to be careful and stop when symptoms occur especially in the beginning but I believe the issue can be avoided. The key is to stay relaxed and focus on minimizing mistakes rather than trying to type very fast from the get-go. Typing with this system is actually not that different from playing a console game.

So I guess the question is, do people who play LOTS of console games (e.g. professional streamers) get RSI?

Some do but as with RSI from keyboard/mouse usage it affects everyone differently. One thing they usually have in common is excessive repetition for example by constantly using shortcuts with CTRL. While practicing typing (both keyboard and gamepad) I noticed that highly repetitive lessons with lots of text like "jf jf jf fj fj fj jfjfjfj" cause much more strain than lessons with more realistic text. In the end it all comes down to responsible usage, nothing is completely safe.