You can actually get it back! I can't take credit for this, but after months of being frustrated about this, I checked to see if someone had found a solution, and I found this Reddit post:
https://www.reddit.com/r/leagueoflegends/comments/4rhymk/what_happened_to_the_spectate_high_elo_players/
Instructions:
Go to C:\Riot Games\League of Legends\RADS\projects\lol_air_client_config_na\releases\0.0.0.58\deploy\ and find the file called "lol.properties" (This is a text file, so open it with a text editor. You will need to adjust your folder path depending on install location or version changes.)
Find this line:
lobbyLandingURL=https://frontpage.na.leagueoflegends.com/$localeCode/client/landing
Change it to:
lobbyLandingURL=
Save the file. (It's also a good idea to save a backup of the original in case you wish to revert easily.)
What the client seems to do is load a webpage onto the main client page. When you remove this URL, the client will show a different layout, and the Spectate Live section will be on the upper left of the main page.
Since this change is just preventing a page from loading (You could also possibly just block this with your firewall instead of altering the file, but I haven't tested this.), I don't believe that this is against Riot's terms of service, and they're still providing the old spectate functionality. This doesn't change the in-game files in any way.
Again, I can't take credit for this, but let your friends know that they can spectate again if they wish!