I have no personal experience with the "Cobweb" antenna but the design is relatively simple. IMHO, you should have better performance on 20-10m bands as long as you raise the antenna at least 35 ft above ground.
Obviously you will lose coverage on 40 and 80 meters. End-fed antennas are nice and easy to deploy but they are not known for superior performance. It's a great compromise antenna for stealthy installations or for quick deployment in field conditions. BTW, I'm sure you know this but it's worth mentioning since your antenna is attached to the kids playhouse. End-fed antennas can develop very high voltages and should be kept from being accidentally touched, especially near the end.
Can you install a ground mounted vertical in your backyard? If yes, look at "Butternut" multi-band verticals. I've had one that covered 80-10 meters. It worked very well, especially for DX due to it's low radiation angle. It will require at least a dozen radials than can be placed on the ground and fixed in place with small plastic clips to keep the wires from being cut with a lawn mover. For ground mounted verticals, the radial's length is not as critical as for above ground applications. I think "Butternut" sells a radials kit with multiple elements cut to optimum length to cover all bands.
"Spiderbeam" is another great choice if you can install a 33 ft mast and have a rotator. The antenna is light enough to be turned with a TV rotator but it performs very well.