Smartthings itself does not show up in the Google Home home screen, just the devices you have integrated into Smartthings (lights, sensors, switches, etc.). This allows you to control your Smartthings integrated devices through Google Home and Google Assistant. I see you have a room labeled 1st Floor with 1st Floor Lights assigned to that room. That should be either your smart bulbs or switches you have in Smartthings, correct? If you scroll down to the bottom of the Google Home home screen there should be a section labeled “In Your Home”. All of your Smartthings devices should be in there. From the Google Home home screen select settings then scroll down to “Rooms, Groups and Devices”. From there you can select the room you want to add devices to and then add them to that room.
I recently added a few new devices to my home and they didn’t immediately show up in my Google Home. I ended up going into the Smartthings Classic app, unchecked the toggle button that @Bry mentioned earlier and then turning it back on. Everything, including my newly added devices, then showed right up.