Once you get to your destination and start walking around or taking public transport you will see how many locals have their phones out ALL the time. In fact, someone recently said "How do you tell a tourist on the Paris Metro?" - Answer - "They are the only ones who don't have their phones out!" Too True!
I also carry mine. I also have a small cross body wallet that I use in transit for my passport and phone. It's the Eddie Bauer Connect wallet but mine is older and is of softer fabric than the models they are making now. (OT - Why do they mess with a design when it works perfectly?) On occasion I have used this for my phone while out and about but it's a pain to either look at the map or rewind if you miss something. Don't leave it lying on a table in a restaurant but walking with it in your hand will be fine.
BTW, if you are going to museums, look ahead of time for free apps. Some are audio, some are just visual. I recently visited Netherlands and extensively used the Rijks app for the Rijksmuseum as well as the one for Mauritshuis in Den Haag. I'm not crazy about the Louvre app but perhaps they will update it.
I'd worry about noise cancelling earphones on a city street just so you can keep track of what's going on around you.