I don't have ready access to US censuses but Lois some years ago gave me a list of Manx born in 1880 US census - in Illinois there are a Thomas Cain + a John Corlett of matching ages to those you quote from manifest - might be worthwhile looking at these in the earliest US census (was there a 1860 ? for Illinois?)
How big was the farm Robert had - can't see his putative family having much money so was it a grant or did he work prior to acquiring land ?
Manx (certainly in Ohio + elsewhere) tended to stick together - were there other Manx born close to him who might give a lead to his background
(I'm assuming that you are 100% certain you have the matching Robert at the American side but it;s the link back to the Island that is difficult

Names of first born child often reflect the paternal grandfather or grandmother - Thomas Arthur Corlett was an important player in Manx affairs in 1770s + some descendants also bearing name but that family was Lezayre based (also a north side parish) - no Arthur I can see in Robert's family tho