Actually AWS scheduled maintenance is the best you can get in the market already, because most of the time they allow you to reboot yourself in order to land an updated host (This time is different might be due to some critical security issues). Other providers like Azure / Google, you have no choice.
(note: I work on GCE, more or less)
Note: I am not saying the 'live migration' in GCE does not work, I am just saying I've more confident (well...peace of mind, IMO) to shutdown the my own database manually (which is automated and tested), ensure all data are flushed to disk, clients are disconnected gracefully and slave has promoted to master.. and things like that, rather than some black magic like 'live migration'
Of course, I am not saying 'live migration' is wrong, but it is just my preference.
There's a story from a partner who was testing it, who at the end of the day said "when are you going to live migrate us?", only to be told that Google had moved them six times that day and they hadn't even noticed.
Alternatively, you have a short period (60s) before the migration occurs during which you can lame a service: https://cloud.google.com/compute/docs/metadata#maintenanceev...
However, I'll grant that both of these lack the "in your own convenient time" portion of your desire.