Times font appearing in Finder Preview

You ended up with a couple of problems that stemmed from removing the spaces in the file names in your font theme. Your original:

<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
<a:fontScheme xmlns:a="http://schemas.openxmlformats.org/drawingml/2006/main" name="AmerType">
  <a:majorFont>
    <a:latin typeface="AmericanTypewriter"/>
    <a:ea typeface=""/>
    <a:cs typeface=""/>
    <a:font script="Jpan" typeface="ＭＳ ゴシック"/>
    <a:font script="Hans" typeface="宋体"/>
    <a:font script="Hant" typeface="新細明體"/>
  </a:majorFont>
  <a:minorFont>
    <a:latin typeface="AvenirNextLTPro-Regular"/>
    <a:ea typeface=""/>
    <a:cs typeface=""/>
    <a:font script="Jpan" typeface="ＭＳ ゴシック"/>
    <a:font script="Hans" typeface="宋体"/>
    <a:font script="Hant" typeface="新細明體"/>
  </a:minorFont>
</a:fontScheme>

Avenir Next LT Pro Regular is not installed with any common software, so it is a third-party font. I don't have it, so instead I used Avenir Light and changed the following two lines, correcting the names:

<a:latin typeface="American Typewriter"/>

<a:latin typeface="Avenir Light"/>

Then PowerPoint saw it as a legit theme. You were lucky with American Typewriter that PowerPoint recognized it without the space. The font theme name should be the name as you see it in the font menu in PowerPoint.

Then when you attach it to your template, you need to go into the master slide and format the text placeholders to use the Theme fonts. Instead of the font name reading American Typewriter, you need to scroll to the very top of the font menu and use American Typewriter (Theme Headings) or Avenir Light (Theme Body). Once you have set these correctly, your template will be instantly updated by changing the theme. Setting the placeholders to plain font names means a theme application doesn't update the fonts.

Here's a link to the updated PowerPoint template, so you can take a look: Themed Template.