From my experience, it's often, but not always that mid is more important early game because of their position to roam either top or bot lane, the fact that their lane is the shortest distance to the enemy nexus and can force team fights and sieges, help their jungler in either their respective north or south side against invasion, help take either dragon or herald or help their jungler invade either camp themselves. However, even though bot can also help with dragon most of the time, bot is more important late game because bot is what will carry a team fight and take objectives like baron with its consistent damage from crit strikes which are not limited to single spells with a high cooldown, whereas the mid takes on more of a support or pick role with its own cc. I usually see more quadrakills and pentakills from adcs than mages, and that usually happens late game.
The reality is though that it also depends on your team's composition and how well everyone's done in lane. If your mid lane is more of a late-game mid laner who wins by scaling with high ap ratios and winning late team fights, but your bot is some early game adc like draven, then bot is more important for maintaining the integrity of your team's progress through different stages of the game since bot will be useless and won't be able to contribute to sieges and team fights when late game arrives unless they have a good early game. So, unless they are against a particularly aggressive early game assassin like leblanc or zed, a late-game mid mage won't generally need ganks unless they are in a lot of trouble from getting ganked themselves or misplaying. This also brings up another point which is that you also want to consider which enemies are more of a problem. If you let leblanc or zed win early game, she can roam and annihilate any other lane, thus allowing the enemy team to take towers. But, if you let some late game mage like Viktor win his lane, you could very easily lose team fights and lose objectives like dragons, so it depends on your team's strategy as well, there's no real one-size-fits-all answer.