Couple of issues here -- please let me know if I'm missing something obvious. First of all, since the HTC One only has the Home and Back buttons, I count on the application to know when to open a Settings screen or even an address bar. Point that browser at YoutTube or Google+ and tell me how you get back to the address bar. I've found that hitting Back may bring up the address bar, but you'd better be quick about it.
Second, the Flash player doesn't seem to work with some sites. I can play a YouTube video with the stock browser, but it won't play an NPR sound clip. Thoughts? Ideas? What obvious thing am I missing?
First, Flash Player isn't supported anymore. If you have it installed through sideloading, it isn't guaranteed to work all the time.
As for the stock browser, have you tried scrolling to the top of the website to get the address bar to show. I don't have any other ideas since I purposely disabled the stock browser and use Chrome instead. Perhaps you should consider the same.
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.