In many communities, more common the farther right one goes on the religious spectrum, the parents are usually involved in a lot of the predate activity mentioned here. In those communities (which are also the communities that have a lot of predating activity), the parents (usually the mother) receive the shidduch offer, ask the questions, call the references, and make preliminary decisions. Those decisions are often based on input from her son or daughter about what type of person he or she wants. If the mother says yes, the shidduch is posed to the son. If he likes the girl's information, they go out.
What benefit is there to having the mother (or father) involved? Certainly there are costs. Anytime there is an intermediary (or agent) we run of the risk of the agent misunderstanding instructions. For example, the son might have been vague about what type of personality he is looking for (assuming personality can be pigeonholed into neat categories). The mother will interpret the son's request to the best of her ability, but that might not be what the son intended. Interpretation always leaves open the possibility of misunderstanding.
Moreover, the mother, human as she is, will impose her biases on his instructions. This is true especially when the instructions are vague. Some mothers might do that consciously ("that girl is no good for MY son!"), while others subconsciously. But almost everyone interprets according to their biases.
These risks are transactions costs, which means that if the mothers were not involved, the guy might have accepted the offer. So why do people still call mothers with shidduch offers? Here are a few possible reasons:
1) The guy gets so many offers that he needs someone to sort through them for him. As the old saying goes, a girl needs an agent, a guy a secretary. She just filters out the obviously bad ones and let's him decide about the better ones. But that argument just restates the problem. Sure he gets a lot of offers, and his mother filters them out, but what about the ones she thinks are no-brainers but he would consider? Is the guy really that busy that he can't filter them out himself?
2) An agent makes things less uncomfortable. Let's say this annoying shadchan calls the guy and pushes him to date this girl. He can tell her he's busy with school or has an event that night, but the woman can bother him and reason away his excuses. If his mother tells the woman that he has school, the most the shadchan can do is ask the mother to tell the son that he can find time. Since the shadchan has to keep talking to the mother, who then has to relay the information to the son and wait for an answer, she'll give up a lot quicker than she would if she only had to call the son directly.
This at least is a benefit to having a parent involved. But does the benefit of not being uncomfortable outweigh the cost of missing out on a good girl? I doubt it. It's not that difficult to tell a shadchan that you don't think the girl or guy is for you. After a while all you have to do is put your foot down. It's much easier than telling a date to his face that you don't want to go out again.
3) Parents are more mature and better at making marriage decisions. Alternatively as a second party they minimize the risk of the child making a bad choice. The first argument is very poor. If people aren't mature enough to make these decisions, they surely are not mature enough to get married. Marriage is filled with difficult decisions.
The second argument is a little better, but is not a sufficient reason to have mothers involved. The parents can merely play a role farther down the road, before they get engaged. That would have the same benefit without the risk of the mother filtering out a good girl.
Overall I fail to see how the benefits outweigh the costs here. Having parents play a role is just another cost on the shidduch system and another reason why we have a crisis.