Why would my dishwasher installer connect it by soldering copper tube to the valve? Cheaper? Less likely to fail? Both?
Now that the valve has gone bad, can I replace the copper tube with a high quality mesh hose? Thanks!

Why would my dishwasher installer connect it by soldering copper tube to the valve?

Who knows. Not measurably cheaper or less likely to fail.

Assuming you are speaking of the soldering the tubing to the stop valve, cut the tubing as close to the valve as you can. Replace the valve and reconnect the tubing with a compression fitting. The braided flexible supply tubes make initial installation easy but you already have the copper supply tube in place. Just cut the tubing and replace the stop valve with a compression type stop valve.

Not sure why it was done that way but, I have never seen copper tubing split under regulated house pressure. My DW 's, 3 since 1974, have been connected to the same copper tubing with compression fittings without any leaks. I recently worked on a house with multi $$$$ in water damage due to a split braided toilet line.