is ‘syncretised’ a modifier or a verb?
Yes. That's what a participle is. :)
More seriously, I think most grammarians today would regard syncretised throughout the class of verbs as a "reduced relative clause":
different forms which have been syncretised throughout the class of verbs.
What does the first clause mean?
. . . we reject an analysis that has gerund and present participle as different forms . . .
Has is an awkward choice of words; understands or regards or represents might be clearer. Profs. Huddleston and Pullum reject the (traditional) understanding that the gerund and the present participle are two different things which by historical accident are written the same way. They claim it is just one thing, and they give it a new name which joins the two old names with a hyphen.
Me, I think the principle is correct and the name is awkward.