How to export ArcGIS feature class to Waze?

I think you are correct that the only way to update Waze is manually using the Waze Map Editor.

To determine whether there is any unadvertised means to apply updates in the way you seek I recommend contacting Waze using their Contact Us page.

You're correct that the only way to export this is through the editor, but the scripts that enable the editor are open to developers. You can add features programmatically using scripts via Tampermonkey and similar add-on script tools. That could add features automatically into the editor, but ultimately you're responsible for reviewing and saving your own work.

I've used it to add and manipulate house numbers, but theoretically it could be used for things like streets and places.

The big thing is you're limited to things you can do manually (you can't edit things above your editor level), as well as governed by limits (adding too many features too quickly can limit your credits). Also it's highly frowned upon (even well-intentioned scripts can have negative consequences), and so local managers may want a tight review of what you're doing. Doing so via the editor can give you a chance to review your edits and adds before committing, which can limit the harm runaway scripts can do.

Waze has a large and vibrant scripting community, and I highly recommend reaching out to them. I'm sure many have had similar ideas already and can get you started.

https://www.waze.com/forum/viewforum.php?f=819