We should be able to let some extensions work on native pages like the new tab page. I use a mouse gesture extension and I can't close the new tab with a gesture.
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The short reason for why WebExtensions can’t access new tab is that it would create a major security risk. The New Tab page is privileged browser code, not web content, so giving an extension access to that page would completely bypass the extension security model and give the extension full access to all of Firefox.
Unfortunately, access in the WebExtension model is pretty binary: you can inject scripts or you can’t. We’d have to create something fairly different to allow “safe” access to New Tab (and/or other in-content pages) for something like mouse gestures. To give an example of what it would take to support this safely, we would need to add core support for a defined set of mouse gestures, then allow an add-on to register commands/actions for those gestures. That’s a lot of work for a relatively niche feature, so it’s not going to happen soon, but it’s the type of approach we’re thinking about for the future.