Actual admin notifications are annoying and take place that should be taken by more important content. An action link could be added to the admin top bar to toggle the admin notices visibility.
I don't find general admin notices annoying. I find it annoying how a lot of plugin/theme developers use them for marketing, especially if they can't be dismissed, and even more especially when the notices show on every page needlessly. However, without these notices, we have no good way to communicate "Settings Saved" or "There was a problem" or anything... and it all just becomes wild west again. I'm voting no on this.
> I find it annoying how a lot of plugin/theme developers use them for marketing
A no-vote from me too
I think it would be better to address the plugin authors who abuse the notifications and/or switch to alternatives that don't have them and/or put together a snippet that disables the lot and/or release said snippet as a plugin :)
I think there actually IS a plugin that removes the dashboard cruft of all the major/popular plugins, though I can't recall what it was called or if it was maybe even a gist I stumbled upon. I'll keep an eye open.
I like the way this plugin deals with it, https://wordpress.org/plugins/wp-notification-center/, unfortunately, it's not 100% accurate and some notifications are still not removed to the Notifications Center. But it's a start.
I have also recently written a very little and simple plugin to do that (https://wordpress.org/plugins/freesoul-gentle-notices/) but I think this functionality should be managed and better provided by the core. My plugin could also be taken, improved, transfered to the core and then removed from the depository if needed. If it's in the core, every plugin and theme authors will test their products and ensure their notices are compatible. Until it remains a plugin, it's not possible to know all possible kind of notices that are introduced by the plugins when they don't respect the standard practices.
I find the idea of https://wordpress.org/plugins/wp-notification-center/ better than the idea of my plugin (less simple but more complete). But I have tested it and doesn't work as expected. If it works I would integrate it in the core instead of my plugin. In any case I would integrate this kind of functionality.
Unfortunately, rogue developers will always find workarounds, no matter what is implemented.