Had the same issue a year ago and now it's back. Only runs with Wi-Fi off and denying connections to api.adguard.com doesn't make a difference either. "Mavericks" doesn't allow installing versions 2.x.x.
Apparently, the culprit is the philosophy of the Adguard developer team resembling that of Adobe: Adguard constantly synchronizes the license to its servers even if the revision is no longer supported and not available for public download. First what I did to check if was the case was that I wiped out my system of every trace of Adguard in Terminal with find. I then restored a clean release unlicensed version of Adguard to see if it would launch but it crashed too. So I removed it by moving every object to Trash and only then that I noticed a log file. I opened it and saw the entry saying smth to the effect that "Error: failed to connect to license server".
Now, dear AG developers, I understand the game but if you no longer offer old Adguard versions and taking into account the fact the like 99% of Mac users are on newer systems and therefore are running your newest releases could you do something to not hamper those users that run older versions that you refuse even to bother checking the licenses of. If you take down the license server then maybe issue a notice or switch them to the offline license mode or smth (or ditch this scheme altogether because it makes your users vulnerable).