Yes, you *can* do this. Believe me, Catrin, I am not coordinated and have real spatial issues. There of lots of things about cycling that I still am not great at, but I *can* use clipless pedals and it was not the worst thing I've had to learn.
Your trainer has a good strategy. It's essentially what I did. I rode around flat streets with one foot clipped in, once I knew I could get on and off the bike with the pedals. Then I started clipping the second foot in and out for short periods, until I thought, "OK, this is stupid, just keep the second foot in." I think this phase lasted 2-3 weeks after the initial day my son took me out at 6 AM on a Sunday to learn.
The riding on the grass thing didn't do it for me. I crashed every time and was worried about riding my skinny tires on grass.