I have found this combination to be by far the best.
I have been running this setup for a several years without any issues. The adapter works way better than some integrated solution. It relocates the weight to anywhere you want. Gives you headphone choices. Works well outdoors too. (I only use it on runs with my phone but I don't see why it wouldn't work on a ride.) And if your headphones die you only need to replace them.
I wrote a long blog post about this setup some time ago but can't find it anymore. Anyway, I found the battery lasted about 4 hours when it was set to full power with those headphones. The adapter is a generic unit so you will find multiple brands. For example I also have a NoiseHush version. Just make sure to get the one with a micro USB charger. The first gen units had this goofy charger cable that you can't easily replace.
Finally, another nice feature is that the play and skip forward and back buttons do work with at least iOS. I found it extra handy when watching Netflix on my iPad while connected to the TV. I could play/pause the movie from it.
As for the headphones, those are my personal favorite. Good volume, good sound, and nice fit options. I have sets of both the retail and refurb and all are still working well after a lot of use.