I would go with a 3-rail fence as a minimum, and I still think you can beat that price by a mile. My 4-rail ranch fence has the rails installed only 6 1/2" apart. I don't know about you, but I can't squeeze through that space. The spacing is so close that it acts as a strong visual barrier as well. Starting from a height of 10" you end up with a total height of 50" rather than starting at the recommended 14" for the first row. This is considerably more privacy than you will get with the 36" picket. The fence is very secure and does not flex even when you climb it or sit on the top rail. It is closely spaced enought that it prevented my Goldern retrivers from going under the fence or between rails. For smaller animals you could add a hot wire or garden pickets under the first rail. It doesn't look stupid at 10" from the ground. You will have to average any terrain differences or a hump in the ground is going to close that space pretty fast. Also, the 16' rail panels span two posts making a level installation much easier. The fence is installed in a woven fashion so that you never have all the joints lining up at a single post. I ended up cutting the bottom off several posts because I hit bedrock. This seemed to work out fine. Take your time during installation to carefully lay out straight and level lines and the installtion will look great, and will be a nice foil for planting, assuming you don't have horses.
A final tip. If you can buy through a dealer or equestrian or ranch supply store, ask for their contractor price (look under "fence materials" in the yellow pages). Master Halco is required to post the full retail price on their web site, but you can only buy from authorized dealers. Believe me, lower prices exist and vendors will bid the job competitively if you do it right. Provide a written specifications sheet with quantities, and bid it to two or more suppliers. If you are self employed put your company logo on the sheet. Go with the best price and be ready to do your own pickup.