I ended up connecting the new dishwasher by cutting the 1/2" PEX supply line downstream of its respective valve, then using a push-fit adapter to transition it to the 3/8" compression inlet of the braided steel supply line that came with the new dishwasher.
For those interested, the configuration shown in the picture I posted uses a Wirsbo AquaPEX system; it is meant for quick installations with no soldering or threaded fittings. A collar is slipped over the end of some PEX tubing, protruding enough to cover the joint between the tubing and the intended fitting (about 1/16"). The collar and tubing are then mechanically expanded using a proprietary tool; once removed from the tool, the widened collar and tubing is connected to the intended fitting and will begin to contract back to the original diameters, creating a compression fit.
The collar can be removed using a utility knife and progressively deeper cuts. I experimented with the end I cut off (i.e. the end connected to the old dishwasher) and found that a heat gun can also be used to temporarily soften the collar and allow me to pry it off the joint; I slipped the collar back over the joint, re-applied the heat gun, and found that it contracted back to its original size and reformed the compression fit.
Thanks to those who took a look and gave this issue some thought. I hope the solution(s) I'm posting can help someone else with a similar issue. Take care, folks.