ok, apologies if this has been answered, but why are you being told no to homebirth, legally, it is your right....unless there are legitimate medical contraindications..if it is a straightforward pregnancy, then HB is just as safe as a hospital birth
also, having an epidural makes it more likely that labour will be slower, that you will require forceps or ventouse. or can even stall labour
your best chance of a positive birth, is a spontaneous , natural labour, remaining upright, active, and able to listen to and work with your body, rather than stuck on a bed, immobile with an epidural, and on a monitor and a drip
many inductions fail due to the mother not being able to stand and move and use gravity to help get baby out
a planned c.s for a first baby will possibly make it harder for you to have a vaginal delivery in the future, and it is major abdominal surgery with its own inherent risks to you and baby.
research and educate yourself about birth, and try not to place it in one of two options, an induction or C.s...there are plenty more options and other methods of pain releif too