they don't let you run in browser contexts normally but they definitely let you set your own new tab page(with webextension privileges) as part of the extension which lets an extension do whatever they want.
see: https://developer.mozilla.org/en-US/docs/Mozilla/Add-ons/Web...
they also have most requested urls as a privilege too so it could be recreated and extended with a little effort.
To get around this limitation, one could set up a personal speed-dial start page.