Svg files for SL360 on the roadmap?

The reason why it glitches has something to do with ... uh ... font coding or something. Some fonts work. Most don't. As your image illustrates, your nice sans-serif font (Myriad?) has been translated, or been interpreted/transmogrified, as a serif font (Times, probably). It's to do with the fact that the code of the font is being bundled into the code for the icons so when exported to SVG there is a ... glitch (I'm not completely au fait with the correct terminology (or actual process) here).

If you convert your text within Illustrator from editable text to outlines (SHIFT+CTRL+O), this will render your text elements as shapes. These will then display as intended once exported in the SVG.

Again, grouping your text shape object as a compound path will create a single shape element that can easily be copied and pasted into Storyline. But the benefit of grouping elements into distinct compound path entities is that it simplifies the vector mathematics that creates SVG files. Less individual objects = less code = smaller file size = fewer things to go awry.

Hope that helps. Apologies for the verbosity.