Yes, the router has something to do with this. Also I believe we are looking at more than one issue that people have been having, and only some of the issues have been documented enough for Apple to address with previous updates.
Not everyone will have the problem that appears unfixed for now, and none of the proposed solutions work.
Granted some people may have had multiple issues and most of the people may have had most of their problems fixed by the latest round of software updates and fixes proposed here.
The problem that has remained has been spontaneous disconnects of Haswell MBA (and possibly other Haswell MB) from certain routers. The disconnects are generally bad enough to require a reboot of the Mac in order to restore wifi connectivity, and disconnects can come in minutes to hours without a noticeable pattern.
Of the 2,000 people who have replied to this thread and half a million views, I would guess that only a fraction of those people are affected by this problem, the rest may be fixed by undergoing the patches and workarounds previously mentioned.
I have resorted to adding a specific base station for the 2013 MacBook Air to use in the office, and it works without failure. Connecting to the existing base station causes disconnects with the 2013 MacBook Air, a problem that my 2011 MBA didn't have and none of my other devices have either.
I don't have any problem connecting the MBA with any other base station I regularly use, which includes a N version of Airport Extreme and an AC Aiport Extreme as well as wifi tethering to iPads/iPhones.
At this point the only suggestion I have is to unplug your router and bring it down to the Genius Bar along with your MBA so they can see the disconnects happen. Since my problematic router is installed in a factory, unplugging it to bring it down isn't an option.
Otherwise, it might be good to report what router brand, model, AND firmware version, along with any settings you may have. It could be a specific chipset and/or setting on the router that is causing the disconnects, and give Apple a clue on where to look to prevent this frustrating issue from happening.