Does the Busker archetype negate the Exemplar archetype

The archetype only negates the bardic performance a normal bard has. However, you still can't use both performances at the same time.

The important thing to note here is that the rulebooks aren't intended to be interpreted by a computer, but by humans.

As an Archetype of the Bard class, Busker changes around things from the Bard class. Exemplar is an archetype of the Brawler class that mixes in some bard abilities. In this case, it's pretty clear they're unrelated and therefore don't interact much. Read the rules on Archetypes to see this intent:

The primary way in which archetypes modify their corresponding base classes is via the use of alternate class features. When a character selects a class, he must normally choose to use the standard class features found in the class’s original source—the exception is if he chooses to adopt an archetype. Each alternate class feature presented in an archetype replaces a specific class feature from its parent class. For example, the flowing monk archetype’s redirection class feature replaces the Stunning Fist feature of the standard monk class.
When an archetype includes multiple alternate class features, a character must take them all—often blocking the character from ever gaining certain standard class features, but replacing them with other options. All other class features of the base class that aren’t mentioned among the alternate class features remain unchanged and are acquired normally when the character reaches the appropriate level, unless noted otherwise. A character who takes an alternate class feature does not count as having the class feature that was replaced for the purposes of meeting any requirements or prerequisites.
A character can take more than one archetype and garner additional alternate class features, but none of the alternate class features can replace or alter the same class feature.

As per this, archetypes only edit the class they're an archetype of, and they don't need to specifically state they don't interfere with any possible archetypes from any other classes.

However, keep in mind that the Busker says:

This ability is treated as bardic performance for the purposes of feats, abilities, and other effects that affect bardic performance. Like a bardic performance, a busker stunt cannot be maintained at the same time as other performance abilities.

So multiclassing between then to try and stack benefits won't work. As a GM I'd also be inclined to encourage a player to make different class choices just to keep down the amount of variables to check.