Since Elementary OS 5.1 is based on Ubuntu 18.04 LTS I would suspect you would have similar issues, if not exactly the same.
There are other flavors of Ubuntu that you might want to consider trying that are on a newer code-base.
You should be able to use 19.04 or 19.10 if so inclined.
Maybe check out "Ubuntu Studio"
This was not an issue for me until the latest update I installed on 01/04/2020. I verified on three different computers. No issue until this update was installed and this was the only update installed.
The "remember-tabs" and "save-exited-tabs" settings make no difference. An extra tab always opens for me.
I do not know how to add it into the menu dropdown, but I have managed to achieve hibernation and set it on the Lid close. Same way if it works for you, you can add it on the powerbutton hit just by edit the appropriate line according to the last title in my answer there - https://elementaryos.stackexchange.com/a/21812/12483