Nefreet
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I would imagine not.
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 (typically the Core Rulebook or the Advanced Player's Guide) - 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.
Emphasis mine. I would interpret that last sentence to mean that there are base class features and there are alternate class features. The Amateur Gunslinger feat references the Gunslinger base class features of Deeds and Grit, and not the alternate class features of, say, the Pistolero.
You could also examine the feat itself:
Although you are not a gunslinger, you have and can use grit.
Prerequisite: You have no levels in a class that has the grit class feature.
Benefit: You gain a small amount of grit and the ability to perform a single 1st-level deed from the gunslinger deed class feature. At the start of the day, you gain 1 grit point, though throughout the day you can gain grit points up to a maximum of your Wisdom modifier (minimum 1). You can regain grit using the rules for the gunslinger's grit class feature (see page 9). You can spend this grit to perform the 1st-level deed you chose upon taking this feat, and any other deed you have gained through feats or magic items.
Special: If you gain levels in a class that grants the grit class feature, you can immediately trade this feat for the Extra Grit feat.
Again, emphasis mine. The feat references the Grit class feature listed within the Gunslinger base class on page 9 of Ultimate Combat. One could easily extrapolate that the feat is referring to the Deeds class feature listed within the Gunslinger base class as well, and not an archetype.