This argument it both boring and trite but I'm bored so I might as well try to educate a poor soul.
if you dont build ANY damage items and someone else builds ALL damage items.. i DON'T think you should EVER win that fight in any champ under ANY circumstance (assuming equal gold)
That is not how a 5v5 MOBA with 120+ Champions works.
The game is not balanced or designed around what would happen if every Champion were trying to 1v1 every other Champion. In the end you have to consider the team element, not to mention other game design elements like objectives, pushing lanes/towers, vision/jungle control, zone control, etc.
Some Carries are designed to eliminate single targets.
Other Carries are designed around AoE or Multi-Target damage.
Yet Other Carries focus on their overall utility rather than raw damage output
Some Tanks are all about protecting an ally through a shield or defensive boosts
Other Tanks are all about soaking up insane amounts of damage and disrupting the entire enemy team through AoE damage or CC
And yet other Tanks focus on their overall utility and being able to separate a single target from their team
And that's assuming that the "All Damage" and "No Damage" Champions you are talking about are in fact Tanks & Carries - which you didn't specify, so maybe you're saying that a Full Damage Braum should be able to defeat Full Tank Vayne?
League has way too many Champions, too many item builds, too many metagame strategies to simply be boiled down as "Damage > No Damage"
If it were that easy, we'd all have left by now.