I completely get why the pile of mail happens. Who enjoys looking at those bills especially if finances are tight and time is limited? I am not suggesting that you look at them anymore than necessary. I am actually going to help you know exactly where they are so that when bill paying time comes you spend less time on that task. (you don't have to waist time looking for it)
The first thing you need to do is purchase/craft/or find some sort of mail holder or box that is large enough to hold all your bills & papers for your current pay period . Make it pretty so you don't mind it being visible. Set it up in a place with very easy access. If you have to dig to find it then you won't really use it. This will be a staging area where you put your mail and papers until the time to deal with them comes. As your mail comes each day then sort through it. Put everything that requires that you pay it, or deal with it in into that holder. Trash the junk mail immediately and enjoy the rest of your mail. This makes mail time a fun time rather than a stressful time and it eliminates that unsightly pile where things tend to get lost.
When the time comes you can find everything in your holder and pay or deal with it as needed. Now file the item in the appropriate file after it is dealt with. As you add more items to the file in the months to come be sure to file everything behind the items already in the file. This keeps everything in the file organized by date. Older items at the front, newer items at the back thus making it easier to locate items (or you can do it in reverse new in front, old in back). Don't get behind on filing by making a pile or chunking it in a drawer instead, because it grows into a monster of filing that you might leave until the end of the year. FILE IT, DON"T PILE IT! File a little bit now and save hours of time later.
A tip: when you pay a bill then make a note on that bill of the check number or authorization number, the date and amount paid. Make note of the authorization number in your checkbook also. Anytime there is question on whether a bill has been paid then the answer is in the checkbook and files.
As the digital age comes upon us more and more, there will be less and less paper to deal with but until that day happens, this is an easy low cost way of staying organized and dealing with that mail.