Yeah, um, NO. In this example, ASCAP is not taking royalties on songs that are not with registered members. They are also not denying funds to those who are. The fees that this particular artist is referring to are quite common, and it is one of the ways that various venues chooses to pay for the fees that they owe for allowing songs to be performed publicly at their venue. A different way to do it would be to simply generate an invoice for each artist who performs a song where a royalty is due and give it to that artist on the night of the performance, but that's not practical, so a lot of places will simply calculate a blanket fee and charge it as one of their terms and conditions in their contract. Notice, this is not ASCAP, this is each individual venue. This is not ASCAP's contract, nor is it their terms.
Also, the artist in the article is being very misleading. The claim is that money is collected from some, but given to others. That's not what's happening. The fact is, if you're a less well known artist, it doesn't seem unreasonable to me that you may have to fill out a form to get your check if your song(s) were performed live somewhere. I don't think it's crazy that ASCAP, or BMI, or any of those bodies doesn't want to get into trying to track every single live performance of every single live song at every single venue, and scour their database in real time to find when/where a song in their catalog has been performed. This person writing the article is basically saying "WAAAHHH!! I wish I made as much money as A-list artists and had people to take care of all this annoying paperwork for me."
So, strike two, champ. You still don't understand how it works, and you remain clearly in the wrong.