The driver uses a 'tinyurl' because dashboard tiles are limited to 1,024 characters, including all the HTML that is used to format those tiles. You are free to put the icons anywhere you like and the driver allows for that. Be aware the character length of the URL you do use may impact the presentation of the tiles. For example the default 'TinyURL' is 27 characters long. The full Github URL that it points to is 72 characters long, or 45 characters longer than the TinyURL. If the tile exceeds 1,024 characters it will not display at all. This driver will attempt to remove icons to shorten the overall length to make it presentable. There can be up to eight icons, depending on the tile, and the URL length you use must be repeated for each one of those. It does add up quickly.
To add this driver's dashboard tiles, see this post.
This driver does not directly feed any other driver's dashboard tiles? If by 'Hubitat dashboard by Joe' you are referring to this, then that is not a Hubitat platform dashboard, it is an Android app. That android app does present the same icons used in this driver (I have it on my android phone and the icons do appear). You would have to get with it's author to try to figure out why they do not show on your Android device.
I'm not sure what you mean by 'HE stock dashboard'? You would have to get the author of the 'HE stock dashboard' if you want them to make changes to that.
Hope that helps.