Exactly. This is my chain of thought on the matter, I'd appreciate seeing where people differ so I can figure out why there's such a disconnect.
1) D&D is both a game and a social exercise.
2) We play D&D for both elements.
3) The point of socializing is to have fun.
4) The point of a game is to challenge ourselves and thus achieve satisfaction from our mastery of the game's required skill set.
5) Proving mastery can only be obtained by comparison to measurable criteria. In D&D, these criteria are encounters.
6) The socializing skill set of D&D is distinct from the "game mechanic" skill set of D&D.
7) The game mechanic skill set has a character building skill subset and a table-play skill subset.
8) One demonstrates mastery of the entire "game mechanic" skill set by overcoming encounters(table-play) with a character you've designed (character building).
9) D&D is a team game.
10) The goal of a team is to prove mastery of the game's skill set.
11) In D&D, the effectiveness of a team is measured by defeating encounters.
12) To demonstrate mastery of the character building skill subset, one should make a character that contributes to the team.
13) The team will be most successful at overcoming encounters the more skill sets the character can provide via his character building.
14) The skill set of a team is not just the sum of individual skill sets, but also the sum of all synergies of overlapping skill sets.
15) Providing more options provides more synergies.
16) Necessary skill sets in D&D are damage, damage mitigation, encounter control, and narrative control. Every ability provides or modifies these base sets.
17) Fighters provide high damage and moderate personal damage mitigation.
18) Wizards provide moderate-to-high damage, high damage mitigation (blur, mirror image, invisibility), high encounter control (fly, invisibility, solid fog), and extremely high narrative control (teleport, scry, planar binding, astral projection).
19) By providing all required skill sets, a wizard provides only benefits to the party.