AI Export not showing font

check the overprint panel.

Here are a couple of tips: first, find out what software the shop that will print the hoodies is using and/or what kinds of graphics files they prefer for their work flow. Not every screen printing shop can handle an Adobe Illustrator file in its latest version. PDF and SVG files can be lots of trouble, depending on the export settings and the application used to export the files.

Generally speaking, an Illustrator AI or EPS file saved down to legacy version 8 compatibility will successfully import into a wide variety of graphics applications (including OLD versions of applications), provided if the artwork doesn't have a bunch of Illustrator-dependent effects baked into it. Flat colors, basic appearance, etc ensures the easiest compatibility with other software.

Also, convert any active text objects to outlines before exporting the file. That especially goes for any text on path effects. Even if the user on the other end has the same exact fonts (same vendor version, same font build) the text on path effects can still be trashed when imported into something other than an Adobe application. As long as the screen printer has all the graphical shapes of the artwork he can change object fills from black to white in just a few seconds. Still, a properly exported file should be able to open in a non-native software environment without any glitches.

There are some steps between the first image and the second one.
1. Which format did you export the artwork to?

2. Why? As far as I know, the printing houses use PDF which does not need any export, just saving directly from Illustrator.

3. What does the exported image look like instantly after exporting, on your machine?

4. The second image is inverted. On which step has it been done?

Now it does not look like any AI issue, more like an error in the pre-print process.

It doesn't look like your objects are disappearing, it looks like it could be because the text is white, they're not showing up on a white background. Why is the final export inverted colors than the first pic?

What kind of file are you exporting and how are you viewing it?