Most of the Janome machines have a speed control that controls how fast you can go even with the "pedal to the metal".
I have 3 granddaughters that I have started on a full size machine. I set the control to "turtle" and they can't go too fast even if they press all the way down!
The oldest was about 8 when she started and her sister was 6. They both worked on a Janome Jem Gold that I had bought for their mom.
The youngest sister was 5 when I started her on my Janome 11000. (Get the feeling I like Janomes?) She just zipped along sewing strips together.
I wouldn't invest in a mini machine. They would outgrow it too fast.