For beans and peas specifically, I usually chit, then plant directly into soil at the recommended depth. Your conditions may vary though, you may have good reasons for starting beans in pots.
I think that generally, any plant grown in a pot should be watered carefully. While the goal is 'just right' watering, a slightly under-watered plant is healthier than an over-watered plant.
FWIW, I grow seedlings in plastic pots only. I tried the paper option, it just didn't work for me. It may work for you though!