Yes. My iPhone 5 is 3 years old and looks perfect and runs perfectly. It's also running iOS 10 with NO problems.
Battery life isn't amazing anymore but with close to 1095 days of charge cycles, that's just going to happen.
If you don't like the music app in iOS 10, you can delete it! There are a lot of 3rd party apps that will play all of the music synced to your phone. You won't have Apple Music if you delete the stock music player but that's a choice you can make.
Just keep looking. You may find a newer phone for the same price. Maybe you have stuff you can sell like video games or clothes. The SE is nice and should be coming down in price soon. But I think you'll be super happy with a 5s
Some of you guys are actually so embarrassing though, acting like the moment something better and bigger than that phone comes out that it's better than all the other phones before it
Some sellers are actually very reliable and sell thousands, even offering a 1 year warranty. They also "grade" the refurb job/cosmetic damage from a grade A downwards and explain what that will mean. For example I once bought a grade A refurb on eBay after I smashed mine, and it seemed essentially brand new.