I'm going to leave this here as I had the same issue. I had to replace my motherboard and then had activation issues with 2 games.
In Windows 10 go to this folder location: C:\ProgramData\Electronic Arts\EA Services\License
EA Help said to delete file "71067.dlf" but that didn't work for me.
After some searching i found the solution here; https://answers.ea.com/t5/Other-EA-Games/Syndicate-Could-Not-Activate/m-p/13582778#M86064
As per the post by Frozenoulis I deleted all the files. The 2 games that where causing me problems then activated without any issues.
Coincidentally one of the games that didn't work for me was Plants vs Zombies. The other was Mass Effect 3.
