Looks like you are off to a great start. Trust me and the others that have already said you have enough coverage and density. Giving up too early is what most guys do that think they do not have what it take to grow a nice beard. The gaps you mentioned are nothing at this point. Give it a full 13 weeks.

About that barber of yours - if he wanted to start trimming your beard with only three weeks of growth, never let him touch your beard. I might give him some slack if he just want to shape your neck line. Barbers without beard experience generally butcher beards.

So now that you should be convienced you can grow a good beard, what are your plans? Do you want a short trimmed beard, a massive beard, or something in-between? I suggest you push your goal beyond what you may think now, and track your progress with photos here. Then you will have a good record of the various lengths, you can trim back to the size you like best. But be warned - you may get hooked and just keep growing.

My Beard Journey:  http://jefffsbeardboard.yuku.com/topic/25137/Finally-I-have-a-Beard
Beard and Bald Thread:  http://jefffsbeardboard.yuku.com/topic/27831/Beard-bald-head-shaved-head-or-buzzcut
Patchy Beard Success Stories:  http://jefffsbeardboard.yuku.com/topic/33742/Patchy-Beard-success-stories-Before-and-After-photos#.ViujbH7nuM8
Taming the Wild Mustache:  http://jefffsbeardboard.yuku.com/topic/30822/Taming-the-Wild-Stache#.Vj_Mh7erTIU