Why doesn't duct tape work well on cables?
To answer your questions:


"Why doesn't duct tape work for cables? Is it the fault of plasticizers? " - No -- it is more likely that exposure to air dries the glue, causing it to eventually lose its tack and become unstuck. This is also the reason that you're often left with sticky residue, as air was unable to penetrate to the underside of the glue.
"Does the sticky film pose a health risk?" - The 3M MSDS does not list any potential health hazards associated with physical contact.
"[...] alternative adhesive tape that works with cables? [...] strong enough to hold cables to a wall." - For simply bundling cables, I like to use the the Velcro cable ties. I do not suggest buying them from general/electronics stores, where a pack of 10 may cost you $5 -- I get mine from Home Depot in the area with the other velcro, which has 2 small rolls of 25 (50 total) for ~$5. 

Tape will never make for much of a semi-permanent wall mounting solution, as it is almost always intended to be a "temporary" solution.
If screws are an option -- I much prefer the use of "Wall Dog" anchor screws (or other anchoring alternative), and eyelet-zip-ties or J-hooks, for securing cables to the wall.
If screws are not an option, you might consider the 3m command strips with zip-ties.