Yes covers support the opening and closing states. They also support a percentage of open. Or maybe it is % closed? I can’t remember which way it is but it annoys 50% of users (who think it should be the other way).
As far as I can tell the “state” of a cover in Home Assistant cannot be “stopped” such as seems available in your Homekit reference. Indeed it can indicate a position in percentages, but that does not affect the state.
I have successfully been able to inverse the icon color of my covers (roller blinds in my case) by using card-mod and setting different colors as follows: