I'm not a professional plumber, but in my DIY opinion, I'd remove what's left of the old, green 1/2" valve and just start over.
In my limited experience, trying to desolder the valve will likely cause the joint on the other side of the wall to come free or leak, so I'd suggest cutting it. Use a pipe cutter, hack saw, oscillating tool, or whatever else ...
It depends on your experience level.
Can you solder?
Do you have pex crimp tools?
If the answer is no to the above two then I'd go for the quick tee adapter. When something in the house breaks that needs a plumber then have them tees and shutoffs in.
The shutoff in the picture is terrible and I always find those leak, if that isn't working anymore another ...
The best way will be a matter of opinion.
In my opinion the best way would be to remove your existing valve and solder a male threaded adapter to the pipe. Then attach a brass threaded tee. Then install two separate valves, one for the sink faucet and one for the icemaker.