I am not an expert on your question as to the Line from Cortland to Eaton, but I think you are referring to the Utica, Chenango & Cortland which was a paper railroad planned to connect Utica via the Utica, Clinton & Binghamton at Smith's Valley (Randallsville) and run down through the Otselic Valley through Georgetown, Otselic, Otselic Center, South Otselic, Cincinnatus and into Cortland. Eventually the fall out of all this became the Cincinnatus Branch of the DL&W with no connection to South Otselic and further up the valley.
Also a line was actually built connecting Georgetown with Georgetown Station on the NYC / Westshore Chenango Branch in the early 1900's. Apparently it was never opened for service and was torn up after a short period. I suspect NYC did not want to take over operation, lease or buy it and freight continued to be drayed from Georgetown Station to Georgetown proper.
You mention Eaton, and I can't recall that as part of the plan, but I also do not know if an actual survey was done and filed. That would be interesting information if anyone could get to Wampsville and check it out. One reason being that I cannot figure how they expected to get from Smith's Valley (Randallsville) up the west side of the Chenango Valley and into the Otselc Valley. Indeed there would have had to have been a crossing with the Chenango Branch somewhere probably between Lebanon and Georgetown Station and with the NY&OM line at Otselic Center, most assuredly as part of the big Rainbow Bridge over the Otselic River.
For years both Georgetown and South Otselic championed rail service and several times it looked like they would certainly succeed but the short-lived Otselic Valley Railroad from Georgetown to Georgetown Station was the most that ever happened. The DL&W was enticed numerous times to extend their Cincinnatus Branch and at one time a grand plan was schemed to introduce interurban service, I think from the Southern Tier near Binghamton up into the Otselic Valley.
Plans were also hatched to have the NYO&W rebuild the DeRuyter Branch and go to South Otselic or Georgetown or both, for the DL&W to assume parts of the DeRuyter Branch out of Norwich, of which they actually made an inspection tour with NYO&W officials over the roads and for an interurban to use parts or all the NY&OM from Norwich to Cortland or beyond over the LV lines to Freeville and connect north to Auburn via Genoa and the shortlived NY&OM / Murdock Grade.
Lot of plans, lots of ideas for the Otselic Valley which kind of dead ends like a box canyon to the north and ideas of connection of Elmira and Cortland to Utica which was a big user of coal for the textile mills and other industry with outlet to Canada and points east.
I don't know how a railroad would connect Eaton with Georgetown Station area such as along NYS Route 26, maybe via a big horsehoe curve out of Eaton and then ? in between the lakes and reservoirs I guess but I don't know the grades and profile that well. However, by Eaton the railroad has climbed out of the Chenango Valley and attained a good elevation.