I have been experiencing this issue as well. It seems to be intermittent. I used to get the reconnect button a few patches ago but now I get the message "Skip waiting for stats"
I think the whole issue started 2 to 3 months ago for me. I have Uninstalled and reinstalled league and still have the same issue. Looking at some other threads I tried repairing my client and making sure that my Video card drivers were up to date (I also tried completely removing them and reinstalled). I have also tried running the game as administrator, but to no avail.
I use Windows 7 for my Operating system and AMD Drivers for my video card. (if possible post your operating system, Processor and what Video drivers you use).
Lastly. I noticed that the issue occurred for me when playing league though the PBE client as well.
I put in a bug report but I have not heard anything back just yet.
I have gotten a few leavebuster warnings because of it and have been leery of playing ranked until i can figure out what is going on. What is funny is that the issue appears to be random but I have noticed that If i can get in one successful game that the rest seem to go without issue most of the time. I even have the issue happen with games I play with the practice tool (I sometimes gauge if I can play without issue if I can successfully get into one of these games).
I feel like there is something that all of us that are experiencing this issue have in common as there are other users that are having this same issue (it could be something that changed with a patch but that makes me wonder why it affects some but not others -> that makes me think it is some common with all of our computers - I want to figure out if there is a way to fix it).
Here are a few recent post noting issues similar to yours:
https://boards.na.leagueoflegends.com/en/c/bug-report/M55IuY7w-cant-play-a-game
https://boards.na.leagueoflegends.com/en/c/bug-report/LG8cBXEl-client-crash-games-no-longer-reload
https://boards.na.leagueoflegends.com/en/c/bug-report/7beTMKYG-game-doesnt-start-goes-straight-to-skip-stats
https://boards.na.leagueoflegends.com/en/c/bug-report/Wn8cAEId-i-now-have-leaverbuster-for-something-on-riots-end
https://boards.na.leagueoflegends.com/en/c/bug-report/0bEBnzNs-game-wont-start
https://boards.na.leagueoflegends.com/en/c/bug-report/ITgIbhq9-connection-error-bug
I hope this gets seen and we can try to figure out is going on.
-Etrai