Thirdly, and this gets a bit more complicated,
@DaleTheWhale kindly linked us the store page for the original skyrim (see below)
Now, clicking that link will open the steam store page in a web browser. However, if you add "steam://openurl/" at the start of that in your web browser, it'll either a) open directly in Steam or b) ask what you want to open it with and steam should be the top choice (or in my case "Steam Client Bootstrapper"), which when you select it will open it up in Steam. Because trying to link that directly causes the forum to remove the fist colon, I can't link it. However i'll paste it as code so you can copy paste it into your browser.
Code:
steam://openurl/https://store.steampowered.com/app/72850/The_Elder_Scrolls_V_Skyrim/
If that works, and opens up the Store page for Skryim LE in Steam, see if there's a "Play" button near the top. If there is, when you click it it'll bring up the install popup you'd normally get. Fingers crossed for you, but I suspect that it'll be a case of waiting on Steam Support.