In my experience it differs from school to school, but since you specifically stated "top schools in engineering" I would totally agree with @aeismail that a "connection" is not needed, but might be helpful.
Also, I have come across the fact that it is common that PhD positions are "created" with a particular candidate in mind, ergo it is very hard to get a position at first try but an application will often get you to a meeting with the prof / research group representative for the department you are interested in, and will dramatically increase the chances that a position might be "created" for you.
Should I apply or just not bother?
You should apply, worst case somebody at least had a look at your resume and you will be higher up on the list for the next PhD recruitment ;)
Do note that top universities have the same issues as many other top tier employers, how do you sift through thousands of applications in a responsible manner? The answer is that you don't; you pick the first and best one, which usually is someone known from previous relations.