Going home might actually be what helps your body go into labour. At home you can also go up and down stairs, go for a walk etc... These things can enciurage labour to start bc your baby's head will press down on your cervix, which releases prostaglandins (helpful labour hormone!) If you go home, you can also have some intimacy (like, massage etc, not intercourse, obv) which increases your levels of oxytocin (another helpful labour hormone). Most importantly, if you go home you can chill out and REST so you'll be fresh when contractions begin
Other things that can encourage labour: nipple stimulation, fresh pineapple, rye bread.
Right now, you probably feel stressed and like you're on a stopwatch. These aren't helpful to labour starting AT ALL. Birth ball bouncing IS good, helps get your baby in a good position and helps release prostaglandins etc... But if you're in that kind of stressed mindset, that can be enough to hold you back, with all the bouncing in the world. A watched pot really doesn't boil.
ETA: Infection is more likely to happen in hospital, but in any case, the vagina isn't a straw.
