Yes, cutting it will help it root. Have you thought about heathers? Would you want to kill off the grass and have a clean blank canvass then to work with? Would you like mixture of shrubs? Do you want something eye catching or something to blend in with surroundings? There are quick growing conifers like juniperus blue carpet and some green varieties that will enjoy poor soil and that slope. Maybe some cascading colourful plants like helianthumums, Arabis, aubrietia, osteospermums. Maybe a couple more cotoneasters. Why not consider all the options.