Raleigh three-speeds were wonderful bikes. I never road anything as late as 1972, but if your bike is anything like the older Raleighs, it should be extremely reliable and a very smooth ride.
You should clean and re-grease all the bearings, including the bottom bracket. It will cost a total of about $3.00 to replace the all the bearings, so you might as well do it when you do the cleaning.
I can't see anything wrong with the brakes from the pics. The bike has traditional side-pull caliper brakes; tried and true. Should be good for what you need. The original brake pads would have been John Bull which are almost impossible to find these days. You can find substitutes an a bike shop.
Keep the trigger shifter. I can't see any reason to change it if it isn't broken.