Me here too, fallen in this rabbit hole decided to invest some time to figure out what is available.
The only home assistant only way of doing this seems to be the platform template_sensor however this has the drawback of you just adding another sensor just to have a dynamic icon, this sounds odd to me.
By using community stuff I found a few other options:
Looks like @Mariusthvdb has made a fork of unmaintained custom-ui stuff, this is quite old, I was looking at the code and seems like the js was minified and now it is formatted again. This relies on setting customization attributes that are later parsed on the UI itself by a js module.
Yet another option is the templatable lovelace card this is a row only component for the entities card, which promises to allow to change the icon.
And last, but not least is the config-template-card which seems to be one of the best options, I ended up using this now.
Are there other options?
Do somebody know why icon_template: isn’t supported on all sensors? Or at least a icon_on:, icon_off: for all binary sensors? Seems quite common specially with the mdi icon library having several version of icons with the “off” state.