fairly classic flex duct insulation wearing out after X number of years. The conditions in the area that the ducts were run in and the age of the materials used has contributed to the failure of the outer wrap / insulation. You have expandable aluminum flex duct. This is far different then the flex duct of today. Here is a link to a product similar to what you have. http://www.novaflex.com/productcart/pc/viewPrd.asp?idcategory=60&idproduct=565
The best thing you could do is change our the flex to steel. This will be more efficient and will last 30 plus years. Yes, Insulation will be hard to do, but in the end you will have a quality installation.
that being said, you stated that you dont want to use steel. so standard plastic flex duct is what your looking at using. as others have suggested, you could used ridgid steel at all elbows and takeoffs etc and use the flex duct for straight runs only. The main issue with the new flex duct is that it must be pulled tight in order to work correctly. This put stress and strain on all the rigid joints that in my experiance, causes the joints to fail/leak. Additionally, there are standards for how much sag the duct can have in a 5 foot run. The percentages of air flow loss are suprisingly large, if the sag is introduced. The best installation method I can suggest from your pictures is to create a supported shelf under the flex duct, thus no sags. Bottom line is that it will move air, how well and for how long depends on the installation.
I would suggest some serious google work on how to size and install flex ducts.
good luck and I hope this helped.