Shuttle cannot be cancelled in '78, not even by Carter - Mondale, nor TPS, nor exploding SSMEs. As long as the Air Force (masking the NRO) plans to transfer KH-9 and KH-11 spysats to the Shuttle, National Security won't allow it to be canned. Neither will Hans Mark, who was seemingly everywhere - NRO, Pentagon Under Secretary of Defense, then NASA deputy administrator.
My bet is 1971. On October 24 of that year is came very, very very close. To make a long story short, Nixon science advisors (PSAC) very nearly convinced the bean counters (OMB) to scrap the Shuttle. That it didn't made any sense economically, which was certainly true.
Whatever Shuttle business case NASA threw at Caspar Weinberger OMB showed minimal savings compared to a similar Titan III driven program. Reusability didn't saved enough, only a handful of $billions that could instantly vanish through cost overruns (C-5 Galaxy, cough, F-111, cough cough - they were NOT encouraging by any mean.).
A rather incredible truth is that back in 1964 when planning Titan III flight rates USAF had done the exact same dumb mistake NASA would make less than a decade later.
They had grossly exagerated future fligh rates and dimensioned the entire Titan III suport infrastructure (from MArtin Marietta to solids to The Cape I T L launch complex) for 50 flights a year or more.
End result: in the early 1970's Titan suffered from massive launch excess capability... and yet, instead of tapping into that, NASA was making the same stupid mistake with the Shuttle. Bad luck for them, this meant that detailed Shuttle economic studies ran head-on into that Titan over-capacity... at the launch market and launch rate levels.
NASA got away with that by artificially inflating the numbers according to the old slogan "build it and they will come". Particularly Spacelab. "Ah sure, the Shuttle will be so cheap at $10 million per flight, everybody will rush for Spacelab missions, dropping launch costs even lower..."
The result of that ? 1990's miseries, Titan IVA vs Shuttle, each $500 million per launch or much more. We ended with two hangar queens and white elephants.
Now, had Titan III-M screwed the Shuttle, things could have been different.
