Let's take a look at the mechanism behind this "checking the quality of your internet connection" (how it is done).
Similar to the common internet speed connection test, you will need two ends to accomplish the "connection quality test". One end is your phone, the other end has to be a server somewhere on the internet. When you establish the Wi-Fi connection with a router, your Wi-Fi program/application in your phone sends a query to the sever asking "how is my connection speed?" If the server returns a value below a certain threshold, then your Wi-Fi program in the phone says "connection speed is too low" and refuses the connection. But if the server is down or unreachable (the network to the server has problems), then you will not get a reply, and so your phone is stuck in the "checking" mode.
I don't know why they think this is a useful feature (this is up to the Samsung developers' wisdom and sometimes it is not so wise). Note that Android is an open platform, cell phone manufactures make their own twist by adding features to it, so there are tons of variations of Android. When the Samsung developers added this feature, they probably think their server is never down / always reachable.
I know this is done by the Samsung developers because my brother has a Google phone which is also an Android phone. One day we stopped by a remote McDonald's which has a slow internet, his phone can connect but not my Samsung Galaxy S3.
Sometimes you can only work with a slow connection, in this case we all know we have to wait for the web pages to download. The Samsung developers might think adding this feature will prevent users thinking their phone is "slow" when connected to a low speed Wi-Fi, but that's a very naive, if not stupid, thinking.
Dont blames on your router, it is a typical "pointing fingers at others" practice which is not always helpful. He even suggested you to download an Wi-Fi analyzer, absurd!! (I don't think I will ever need to analyze a McDonald's Wi-Fi connection.)
I will urge the Samsung developers to drop this useless feature, or at least PROVIDE AN OPTION for users to turn it on/off.
On the other hand, you can always switch to a non-Samsung phone which is what I am going to do