Your '73 Schwinn "10-speed", unless it's been heavily and recently upgraded, has a total of 10 speeds, i.e. 5 cogs in the rear and two chainrings in front. In modern terminology "10-speed" means 10 cogs in the rear alone which, combined with two or three chainrings, give a total of 20 or 30 "speeds".

So, those barend shifters will only work if you use them in "friction" mode and they are overkill for your bike. Try to find a set of older, less expensive barends which can be lebeled as 7,8 or 9-speed (all refering to the number of rear cogs) and all of these have the needed friction mode you need.

Edit: Apparently Fastjake is faster at typing than I am but note that we both told you the same things.