We've never carried the wave phones, just the app, and it has always been fine. but I suspect it depends on how old your kids are, and how comfortable they are being dropped. Our bigger kid loves the club, so it's never been an issue for him. our little one doesn't, but we didn't use the nursery often when she was 2, and when she was big enough to be in the club with her brother, she just found him when upset. Anyway, short version - the club has never needed to call us, so we've never had the wave phone. And the CMs have always said it's fine to just have the app.
We've also never had a big problem with the app - the message notifications/alerts sometimes are late, but if you open and refresh the app they show up. So if you are waiting to hear from someone, just check every few minutes if you think there's a delay. Otherwise, you'll get it on vacation time.