Compression valves like the one you linked will work fine on naked copper, but are useless on threaded pipe.

The major issue will be getting the old valves removed without snapping anything off. If the pipes coming out of the wall are galvanized iron they are a ticking time bomb, but the pipes they are connected to would most likely also be galvanized iron. Ergo, wrenching on those valves could snap off pipe in the wall. You'll have to be very careful, and even that may not be good enough.

Back in the day, I helped out an old man whose bathroom shutoff valves were leaking at the galvanized threads. Was going to try to tighten the valves onto the pipe, but as soon as I bumped the cold valve with a pipe wrench it shot off like a rocket. Water was everywhere in an instant. Oops. Learned that day to shut off water before sneezing on galvanized pipe. It was 1/2" galvanized pipe. Luckily I had some 5/8" fittings which fit well enough into the remaining threads to stop the leak. Considered myself very lucky that the pipe didn't break in the wall while installing.

If I'm seeing things wrong and that's brass coming out of the wall, removing the old valve may not risk snapping the pipe off at the threads going into the valve. However, that piece of pipe is threaded into something and odds are very high that the pipe within the wall is galvanized. I simply cannot tell.

Any surgery involving old galvanized pipe risks opening a pandora's box of plumbing issues. Attempting repair of one bit can lead to the next bit farther back breaking. Attempting to fix that next bit can break the next one back.

I totally get your determination to remove that jackass bandaid job, as redundant nonfunctional valves are stupid and the whole thing looks like shit. I'd definitely handle such an issue myself, as I have all the tools and skills to handle the repair and deal with any resulting damage. However, I fear you may lack a few necessary tools and skills. I encourage getting the tools and earning the skills rather than calling a price gouging vulture in nearly all cases, but this particular mission carries a lot of extra risk and may not be the ideal situation for breaking your plumbing cherry. Adding a valve for the drinking water and dishwasher is much simpler and less prone to causing major issues, albeit another jackass bandaid of a repair.

If you must do this surgery, secure the pipe coming out of the wall with a pipe wrench to make sure the only thing that turns is the old valve you are removing. The new valve must have the appropriate npt threads to fit the old pipe. I'm guessing its 1/2" npt, but it could be 3/4" or an oddball size. Assuming it is 1/2" npt, I would use this type of valve to accommodate the sink and dishwasher/ fridge and add a stop tee on the cold side for drinking water. Obviously longer lines or extensions would be needed to connect the faucet.