While it's true that companies love go-getters, that's not really what you're doing here.
The first thing to be aware of is that unless you are talking to someone in your report chain about the job, especially if it is an job you are applying to in a formal method(ie through job listing without a known contact inside the company), then anything you send will be looked at by someone in HR and will not probably get to someone in the chain who would care about the work and effort you put in. That HR person probably can't speak to the quality of your submission to this particular portion of the job. To them you're someone who sent a bunch of 'extra stuff' along with the resume. If you're working with a recruiter, if you know someone in the hiring chain of command then maybe it would be worth it. For something going to an HR front-line worker this is probably going to look weird and probably not in a positive way. You will be the applicant that send a bunch of extra, unasked for stuff. Stuff that's only valuable if you are in the group that requested it. Stuff that only makes sense in that situation.
The second thing to consider is... well there's no easy way to put it. Are you sure that you can do this task on your own, without inside information, without guidance or leadership in a way that is positive and professional(and that the company is looking for)? Heck I'm not in a place in my career where I am looking for internships and I certainly wouldn't want my first introduction to a company to be a cold generated review of software without knowing the context and expectations. You mention that they are explicit in the details but, let's be honest here, that means they probably have a very specific business use case in mind. A general comparison on a wide field of softwares(even if it follows the blurb they put in their job advert) is something that could go pretty wrong pretty fast honestly. It could really make you out to be a much worse candidate than you are. If I were in a hiring position and someone in your position did that I would be a bit put off by the candidate. It's hard to put into words why, I think it's something about a internship/entry level candidate that would be so presumptuous as to attempt to make recommendations from outside the company before meeting anyone in the company. Imagine if someone came into your house and by way of introduction said "That's a decent couch but I think a purple couch would be much better. By the way have you tried the cheese at the grocery store. The Swiss is ok but the Colby Jack is full of flavor and robust!"
The final thing, in my mind, to consider is that if you're willing to do the work for free why should they hire you? Your time, even when looking for a job, is valuable. Doing free work for a company smacks of desperation and devalues your work.
All of this, on reading back, comes across a bit negative. There is a place for what you are talking about. That place is in the interview. Bring up the fact that you saw the specified job duties. Feel out the conversation about those duties and, if appropriate make general, positive recommendations.
Random, outsider attempts at job duties don't belong on the application.