Can a gambrel roof have outer instead of inner supports?
Trying to brace the roof from the outside as you illustrated will not add any significant structural reinforcement. Without the metal bands, the front and back walls carry the brunt of the lateral load that the roof places on the side walls. Adding straps that encircle the exterior will primarily reinforce the ends, and do little towards stabilizing the rest of the side walls.

If you're working on a small scale, such as an 8x10' (maaaaybe 12x12') shed, you may be able to get away with this type of design, where the exterior walls serve as the bottom truss members to tie the roof together over a short span:



(image source)

However, if you're dealing with anything much larger, this method will not be sufficient. The reason is easier to understand when you examine how the different members in a truss work together to support a load. The top members of a truss supporting a vertical load are generally in compression, with the bottom in tension. Without the bottom members pulling your truss back together, the walls of your structure will be forced outward.

Here's an example showing exactly how each member in the generic gambrel truss works:



In order to analyze the tension and compression in each truss member, the standard practice is to assume the joints do not support any torque loading (aka moment in engineering terms). Therefore, each member is either in tension or compression. For the beginner, I always recommend arbitrarily assigning each member in tension, and if it solves to a negative value, that means it's actually in compression.

First, start by determining the reactions at the supports. Since this loading is symmetrical, it can be assumed that the vertical reactions are equal. Sum the forces in the vertical direction, and solve for the reactions:



Next, you can use the section method to determine the tension in member CI. Start by 'cutting' the truss across the members you want to solve for. Careful though, you can only solve three at once - sum of the forces in x and y, and sum of the moments for the resulting section:





In this case I only solved for the tension in member CI, which is what you're interested in eliminating. It came out to 2.1875 kN, which is approximately 1/4 of the loading not applied directly over the walls. So if you calculated the total weight of your roof material, you should be able to get a rough estimate of how much lateral force is going to be on the walls. THEN you could work out a beam bending equation for the top plate running along each side wall, and figure out if it can sustain that lateral load.

DISCLAIMER: I'm not giving you professional engineering advice, I'm just trying to help you understand how truss structures work so you can make an informed decision yourself.

If you want to look more into the methods I used, a Google search for 'statics truss examples (PDF)' may be the best place to get started.

EDIT: I was at Home Depot last night, and noticed they had several pre-built sheds with gambrel-style roofs. I looked inside, and most of them simply had plywood gussets holding the truss members together, with no cross-beams. Although, the larger ones usually only had half of the roof built this way, with a loft to stabilize the roof at the rear half. I'd suggest that you go to Home Depot or Lowes and take a look at their pre-built sheds. Try to find one the same dimensions as yours, and pay very close attention to how it's constructed. I'd even take a tape measure, sketch pad, and camera along and pretty much copy the design. Someone took time making sure that shed is stable, I guarantee it. You may as well take advantage of it for your own shed.