The lawyer isn't happy because in Missouri, if you merely pay a traffic ticket, you are also pleading guilty to the offense on the ticket. In this case, by paying his ticket, the driver is pleading guilty for failing to yield at the intersection. That guilty plea can be used against him in court. Sure, the driver can still get on the stand and say that he was not the one who failed to yield at the intersection, but you can bet that the other driver's lawyer is going to pull out the traffic court file and show the jury that the driver has already admitted that he failed to yield at the intersection in a court of law. This is a very tough hurdle for any attorney to overcome.
If you are involved in an automobile accident and you are issued a traffic ticket, DO NOT JUST PAY THE TICKET. Consult an attorney before doing anything. If you are not injured, you may be able to find a reasonably-priced traffic attorney who can try to get the ticket amended so that if you ever get sued, you can better defend yourself. If you are injured, oftentimes a personal injury attorney is happy to help you take care of the traffic ticket while also handling your injury case.