Because the National School Society was a Church of England Society it insisted that children come into school on Sunday mornings for religious instruction and attend the parish church twice during the day.
Parents who went to chapel had to decide whether to keep their children away from school or let them attend a church which they did not agree with. The teachers at Machynlleth National School seem to have been sympathetic as they let the chapel children off church and the Sunday religious instruction.

More rules for the new school...