in this case they are both irregular

A little more research shows that in OE, settan seems to have been regular. We perhaps think of it as being irregular because it isn't set, setted, setted but it seems the -ed got elided into the final t of set. I suspect the same of put but my AHD and my Concise Anglo-Saxon Dictionary are at odds as to the mere existence of the word putian, and it doesn't speak to the issue at hand anyway, so fuhgeddaboudit.