Haha, 2020 and I still have that problem with my old MacBook white Late2008. It is not the converter (already changed it) and also seems not any cable. Why I think that? Because…
If I delete MacOS and install Windows, or Linux. The screen is never turning off. If I have run MacOS it does, even MacOS is installed beside another OS it starts to sometimes turn off in the other OS as well.
Did someone else had that problem?
Is there a software from Apple to destroy your old device? To me it looks like that.