I have a similar issue, but I do not need to restart my computer. The "solution" I found was to use the task manager to close League entirely (all instances that show up, (which is usually two clients and another 3 windows for some reason......), then relaunch the game. As soon as I log back in it puts me back into the loading screen.
Solution is in quotations however, because doing this apparently temporarily resets your client, basically forgetting you ever played the game. It reenables newbie tips, walks you through the "new 2019 season", and clears all your rune pages. Leaving the total number of rune pages available, but they no longer exist until you remake them.
There is a 50/50 chance if you log out and log back in that the runes will come back. I have had to remake all 21 of my rune pages on 6 separate occasions this week, while about another 4 times I relogged and they were back.
Not sure if this is much help to you guys, but this is the only way I have found to "fix" the issue. Although it often happens again 2-3 games later.