The timestamp device_class for sensors is not supported in all the built in lovelace cards or badges. Last I checked, it was only supported in entities cards.
For a little background, when you place a timestamp device class in an entities card, it will show up as a nicely formatted string. E.g. 5 minutes ago, in 5 minutes, etc. The technology behind this is used for displaying timestamps in secondary info.
The issue is: It’s not used everywhere. If you were to place a timestamp sensor into the entities card, it will display the raw timestamp. E.g. 2020-9-10T12:00:00.00000+00:00.