Tried the loose basket thing for two weeks and it was a total mess. My cat found the pile behind the TV and turned it into a toy, and one of my USB cables got chewed so bad it stopped charging my phone. Plus every time I slid the entertainment center out to vacuum, cables would snag on the legs and yank the power strip half out of the wall. Zip ties at least keep everything where I put it, and if you leave a little slack loop at each device instead of yanking them tight, you can still pull one cord free. Swapping my monitor arm took me four minutes last time, not twenty, because I labeled the ties with tape. Sounds like the real problem was wrapping them like you were shipping a package.

Heat shrink on the charger brick is smart, I tried the same thing once and melted a chunk of the cord because I got too close with the hair dryer, so now my repair job has its own little scar. That said the tubing itself never moved, unlike the tape that always turned into a sticky mess in my bag. Takes a bit of practice but it's worth the extra minute.