I cannot say I know the reference to “The Horseshoe Road Inn” but a little Googling turned up these references. One is a YouTube Video of “Ryan Michael Galloway” singing a song about an ex-wife and the song is titled, “The Horseshoe Road Inn”
Perhaps Galloway or Tom and/or Ray heard the expression for the other or maybe it’s an old expression that goes way back and both Galloway and the Magliozzis picked up it and used it independently.
However, if you read the following Credits that Ray would recite at the end of each broadcast, the expression is at the end…
The ending credits of the show started with thanks to the colorfully nicknamed actual staffers: producer Doug “the subway fugitive, not a slave to fashion, bongo boy frogman” Berman;[23] “John ‘Bugsy’ Lawlor, just back from the …” every week a different eating event with rhyming foodstuff names; David “Calves of Belleville” Greene;[24] Catherine “Frau Blücher” Fenollosa, whose name caused a horse to neigh and gallop (an allusion to a running gag in the movie Young Frankenstein);[25] and Carly “High Voltage” Nix,[26] among others. Following the real staff was a lengthy list of pun-filled fictional staffers and sponsors such as statistician Marge Innovera (“margin of error”), customer care representative Haywood Jabuzoff (“Hey, would ya buzz off”), meteorologist Claudio Vernight (“cloudy overnight”), optometric firm C. F. Eye Care (“see if I care”), Russian chauffeur Picov Andropov (“pick up and drop off”), Leo Tolstoy biographer Warren Peace (“War and Peace”), hygiene officer and chief of the Tokyo office Oteka Shawa (“oh, take a shower”), Swedish snowboard instructor Soren Derkeister (“sore in the keister”), law firm Dewey, Cheetham & Howe (“Do we cheat 'em? And how!”), Greek tailor Euripides Eumenades (“You rip-a these, you mend-a these”), cloakroom attendant Mahatma Coate (“My hat, my coat”), seat cushion tester Mike Easter (my keister) and many, many others, usually concluding with Erasmus B. Dragon (“Her ass must be draggin’”), whose job title varied, but who was often said to be head of the show’s working mothers’ support group.[27] They sometimes advised that “our chief counsel from the law firm of Dewey, Cheetham, & Howe is Hugh Louis Dewey, known to a group of people in Harvard Square as Huey Louie Dewey.” Huey, Louie, and Dewey were the juvenile nephews being raised by Donald Duck in Walt Disney’s Comics and Stories. Guest accommodations were provided by The Horseshoe Road Inn (“the horse you rode in”).
As “someone” else would say, “That’s all Folks…”