Adverts have no place in the admin area. A lot of the clutter comes from plugins trying to sell something, whether it's upselling a Premium version / paid upgrade, or a related product, or worst of all an unrelated advert. They aren't notifications that an admin needs to see, ever.

I realise a lot of plugin authors depend on upselling to make any sort of income from their plugins. The problem with the current system is that it's a noise arms race: if one plugin is loud and another is subtle, the subtle one will get less money despite being the better admin experience, so plugin authors have no incentive to play nice.

So while we need a solution to notifications (such as the WP Notify project already mentioned), we also need a separate solution - not part of that notifications area - for how plugin authors can upsell responsibly. I don't know what that looks like, but without it the notifications area won't solve the problem.