Is replacing the SATA cable to the hard drive and motherboard just a matter of powering off the computer, unplugging the old one and switching for the new one?
In other words, will there be any programming involved, driver installs, etc.?
It's just hardware so your assumption is correct. There is no programming or even reinstallation of drivers.
Power off, unplug the old and insert the new. Make sure that the cables have the same type plug at each end (straight or bent).
Make sure that you and the computer case are grounded to avoid the (small) risk of static shock.