If you have multiple accounts, try signing into another then going back. I don't know why (maybe it caches stuff for your most recent login), but that fixed it. Last night, I got everything on TF's act on my ARAM account (except actually getting the icon, since they said it could take time). I start it for my main account, but tried to finish it today. At around 31 assists, my matches stopped counting toward them -- I was even getting a delay in the Match History, where 3-4 matches didn't show up, and completing a match brought one of the missing matches up instead of the recently completed one. But, even when those matches showed up, they didn't add to the Assist progress.
Resetting the client and logging back in didn't help. But, after logging back into my ARAM account, it gave me the TF icon that I earned last night. Then, I logged back into my main account (this one). I went to the progress page and it didn't update, but when I went to my profile to check the matches, it popped up that I earned the icon. The match history still is behind, but I went back to the event page and it has everything completed.
TL;DR: Try signing into a different account and signing back in and re-checking your progress. There could be a delay on it, so if it still isn't updated, try waiting a minute or two (or visiting another page and returning).
Side Note: ARAM and Bot matches do count. ARAM and Beginner Bots was literally all I played.