I don't have a strong opinion on the interrobang either way, but let's imagine we wanted to start a campaign for it to go mainstream.

The most important thing that could happen would be for a popular auto-correct tool to start substituting it for '?!' and '!?'. MS Word would be ideal.

People would start to get familiar with it when reading other's works. And when authoring, people tend to treat auto-correct like an authority and learn from it. I suspect something similar happened with the ellipsis character in place of three periods.

I have a hunch it would catch on easier in the non-English markets, which may be slightly more accustomed to the inconvenience of using glyphs that aren't always available on every keyboard.

Next, it needs to be included in mobile keyboards. I don't know about iOS, but I can't find it in Gboard for Android.

We could start logging requests and submitting pull requests along those lines.

Now we may not want a viral event. No hashtag campaigns, lest we awaken a louder dissenting crowd. I think the Trojan Horse is a better strategy at first.

I don't know if you've ever played with a font like FiraCode, but it gives you both fancy typographical symbols and compatibility with existing compilers by defining purely-visual translations between certain sequences of characters and purpose-built glyphs. So for example '!=' is rendered as '≠', '=>' as '⇒', '>=' as '≥', etc.

See https://github.com/tonsky/FiraCode. They do go a bit overboard, IMO.