The correct synonym for the majority of them

The reason it sounds strange is that it should be “the greater number of them.” Traditionally, superlatives like greatest are used when comparing three or more things, and comparatives like greater are used when comparing two things. Here, you are presumably comparing the ones who are with the ones who aren't, so greater is the proper word to use.

This “rule” is obeyed less today than it was during the 19th century, but some people still follow it. And maybe you should think of “the greater number of them” as a frozen form, where greater is more idiomatic than greatest.

See Google Ngrams.