The computer does automatically raise shields in several TOS episodes and in early TNG. The reason this is not allowed to happen more often might be directly tied to the reason shields are not being kept up by default.
If it's not power consumption, what could it be?
- Unthreatening posture? This is a smart initial move in a universe where most opponents have divine powers and could smite the starship if angered, regardless of whether shields are up or down (that is, the TOS universe). It is also a smart move in a universe where most opponents have learned that trading of blows will quickly escalate into a war they can't afford, and that a Starfleet vessel with shields down is offering them an honorable way out of the week's dilemma, and not
being either suicidal or insulting (that is, the TNG universe).
- Sensor troubles? If shields prevent accurate sensing, then the better way to prevent the enemy from sucker-punching you is to keep shields down and see him before he sees you. We don't have much in the way of dialogue to support the idea that shields hamper combat, but it's a possibility.
- Stealth? Shields sound like something that would glow on the enemy's sensors more brightly than the local star, whereas we know that it's basically impossible to spot starships in space if they refrain from activating their high energy systems. Indeed, it might be that raising shields makes it significantly easier for the enemy to target you, except at point blank ranges - so it's sensible to keep shields down even after spotting a distant enemy on approach.