Yes, there are good reasons, which might not apply in your department (and somewhat apply to mine), as to why graduate admissions are handled that way.
There are still many places in which admissions, even graduate admissions, are non-competitive. This means there are some standards, which vaguely correspond to having sufficient qualifications to indicate the applicant is likely to be able to complete the program in a reasonable amount of time, and everyone over that standard is admitted. Applications are not meant to be compared against each other but rather only to a fixed standard (even if that fixed standard can't be precisely specified or easily explained to an outsider). When admissions is non-competitive, the system you are describing means that applicants can get a decision on their application fairly soon after their application is complete, instead of waiting until after a fixed application deadline date. It also means that you don't have to have a deadline for applications, since you decide on an application without worrying a better one will come later.
Not having deadlines for applications can be an advantage, since some people don't decide until very late to apply to graduate school. Being quick is also an advantage, because some people may take the first offer they get. Note both these advantages are more important for applicants to the lower-ranked schools for which admissions tend to be less competitive.
You might be shocked that graduate admissions could ever be non-competitive, but this makes sense in several kinds of situations. One situation is for a program that usually barely gets enough qualified applicants to fill their spots. In that case, it makes sense to simply take everyone who is qualified, and the quicker the better, since one is worried some spots might go unfilled. Another situation is for departments which have enough faculty resources to support all their applicants (or whose universities want the faculty to stretch if necessary) in subjects where students commonly go to graduate school without funding. In that case, admissions can be made on a non-competitive basis with funding (which is limited and hence has to be competitive) decided later, even perhaps with students being forced to decide whether to attend or not before they know if they will get funding or have to pay from their own pocket.
Now as to whether you can change this, it depends. First, if in fact your department really is in a situation where it should just admit every qualified person who applies, there is no point in changing this. If that's not the case, and your department is in control of its admissions process, it's not so hard to change.
However, in many cases, graduate admissions is controlled university-wide, and your university may very well have a general philosophy of accepting every qualified applicant. If non-competitive applications make sense for most of the departments in your university and your department is an exception, it doesn't make sense to change the university rules just for you. In that case, your department might be able to get permission to institute a pre-application process where potential applicants who don't first make contact with your department before a certain deadline automatically get rejected, and you notify the pre-applicants soon after the deadline whether it will be worthwhile for them to make an official application (which costs an application fee) or not.
Be careful. If you make a change that causes enrollment in your program to drop, you might get in trouble and your graduate program might simply get eliminated. It's hard enough already for us to explain to our administration why we can't have 30 students in our program with only 15 university-funded TAships, since mathematics is one of the few subjects with neither grant funding that covers tuition for a neither percentage of students nor many students willing to pay for graduate school from their own pocket. If your graduate program runs at a financial loss, your university may not be willing (or even may not be able) to keep it going.