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Having top or bottom solo opens that champion up to a huge world of hurt. If the opposing team has a strong lane controller (Janna, Soraka, Nidalee, etc.), they're likely to end up getting less XP and gold than they would soloing mid. I've faced a solo Ashe as Janna, and single-handedly prevented her from never leaving tower range, while my lane partner got free last hits. That wouldn't happen solo mid, as you can be within range (or flash-range) of the tower and still be getting XP and gold.
Also, the travel distance to mid is less than it is to the other lanes, so it's safer to leave the lane without the tower getting pushed down. Soloing top or bot, leaving the lane means that by the time you get back (barring Teleport), the tower will be almost down.
Getting somebody else to cover your solo lane while you heal has the same issue as soloing top or bottom in the first place, when it comes to zone control. It's really hard for a champion coming from a duo mid to cover solo top/bottom because they'll be the same level as their opponent in a larger lane! They'll be extremely vulnerable to a gank or harassment until you get back. Covering a solo mid doesn't have even close to as much danger, since the distance between towers (and lack of really close brush) is lower.
And did I mention the brush? Hello free zone control against a solo top/bottom.