Initiative comes from D&D's wargaming roots, where initiative was re-rolled every round. Its original purpose was to add unpredictability. For example, the AD&D 2nd edition Dungeon Master's Guide says:
Initiative is not set, but changes from round to round—combat being an uncertain thing, at best. A character never knows for certain if he will get to act before another.
Initiative is normally determined with a single roll for each side in a conflict. This tells whether all the members of the group get to act before or after those of the other side.
D&D 3e changed from group initiative to individual initiative by default. The most likely reason, in my opinion, is to prevent combat from becoming "swingy", where one side acts all at once and can severely weaken the enemy before they can react.
D&D 3e also changed initiative so it was rolled once at the start of combat, instead of every round. This is probably to cut down on dice rolling - individual initiative takes more rolls than group initiative, so it would slow down play to roll it every round.
This partly defeats the original purpose of initiative, which was to add unpredictability to each round. However, it still decides in a fair manner who moves first, gives an advantage to characters with higher Dexterity, and keeps combat from becoming either swingy or predictable.
In my experience, fighters in D&D 3.5 or Pathfinder actually do tend to move first. Their high number of bonus feats sees many fighters take Improved Initiative early on, which gives +4 to Initiative. Clerics tend to wear heavy armour and have low Dexterity, so on average they don't go first.
If the wrong person does go first, he can always delay his action until later in initiative order, or do something useful like cast a beneficial spell or draw a weapon.