But you do have a good point. What about ASCAP and BMI who do regularly sue bars and other venues that play music without paying royalties? They do not require copyright assignment.
Because they are not concerned with copyright infringement. A public performance of a song is *not* copyright infringement.
Under copyright law, you do not have to negotiate an agreement for royalties in order to use a work in a "public performance". Instead, the law specifies a "statutory royalty" that is due whenever a work is used in a public performance, with the rate of this royalty being set by the government (the Copyright Royalty Board). You may, if you wish, try to negotiate with them for a contractual royalty rate different from the statutory rate, but unless you have done so, you are obligated to pay the statutory rates.
ASCAP and BMI are just agents responsible for collecting the portion of those statutory royalties owed to the songwriters/publishers of songs used in public performances. They aren't suing because you infringed copyright (you didn't) - they are suing because in using the work in a public performance you've incurred statutory royalties, which they are (in theory) authorized to collect and fairly distribute it.
(The portion of the statutory royalties due to the artists/labels is collected by an organization called Sound Exchange).
In summary - to sue over copyright infringement, you need to be the holder of the rights to the work in question. But to sue over the statutory royalties for a public performance, you only need to be the agent authorized to collect those royalties.