Why is it embarrassing? That's entirely in your mind. It is objectively quite neutral. It's not about the cost of the phone... it's about the massive effort that goes into setting up a new phone, literally multiple weeks of effort. Also, life can be disrupted a fair bit without a phone. There is also some data loss of apps that haven't fully been backed up. If your phone use is light and ordinary, you don't need it.
I would find a lanyard attached to my belt or around my neck awkward I think. I do use a wrist lanyard through--not som much for this purpose but because I worry that I'll be taking a photo from a bridge and someone will jostle me or I'll otherwise drop the phone in some particularly inconvenient location.
I keep it just as I keep my keychain, hooked in a belt loop. I guess you will understand only after you have forgotten your phone somewhere. When you do, I hope you get it back.
