Would the Saturn-INT 20 be possible as something to come in lieu of the Shuttle, or possibly alongside of it?
Alongside it? Probably not. Any Shuttle design was going to be two-stage-to-orbit, and, no matter what first stage is used (reusable flyback, S-IC, or the OTL solids) it would always make more sense to replace the Orbiter with a disposable cargo pod if you really need 80-100 tonnes-to-LEO in one launch, than to maintain the infrastructure for producing INT-20.
An analogue to Shuttle-C/Z is more likely. Those side-mount cargo launchers were designed to achieve heavy lift with minimal cost--and to use the existing launch facilities to the greatest possible extent. When SEI was proposed, that would have allowed NASA to have heavy-lift
and operate the Shuttle Orbiters at the same time. You could conceivably make an analogous design with other first stages--a reusable flyback first stage could launch a disposable cargo pod from its back instead of the Orbiter. A Saturn-Shuttle could launch a Shuttle-C/Z instead of an orbiter.
Of course, the thread title asks for mundanity. Here's two mundane scenarios:
No Skylab-B, no Shuttle. Titan III-M launches carry Apollo CSMs and CM/LM-derived laboratory modules up into LEO for missions lasting anywhere from a week to a month, but nothing continuous. Eventually, an American Mir goes up, as an answer to the Russian original, with a crew of five, remaining up into the twenty-first century. Titan III-M is replaced with an *EELV, and the Apollo CSM is eventually made modular as the Soyuz is--parts are ripped out after landing to reduce the costs of the construction of the next capsule. While a station is up for a long time, the political support and funding for a proper lunar return or anything more ambitious never materialize. Essentially, America's space program more closely follows the historical path of Russia's after Apollo.
Scenario 2: John Glenn gets into orbit before Yuri Gagarin. Apollo and the early orbiting laboratory program go into service before the 1960s are out, and a lunar orbital mission is flown in 1970. However, the economic downturn of that decade means that no funding for a lunar lander ever materializes. The USAF eventually decides that crewed recon platforms are more trouble than they're worth, and abandons the project. As the space station program never delivers the scientific and technological 'spin-offs' it promised, or in any event not obviously (no miracle drugs, no semiconductor factories in LEO), eventually that program too comes to a halt. Even with reduced funding, the Jet Propulsion Lab lands unmanned spacecraft on Mars and the Moon, even performing unmanned sample return from the latter body. The scientific community appreciates these samples, but without a Jack Schmitt or Dave Scott up there showing what a properly-equipped geologist can accomplish, the case for human exploration does not resonate as well in this timeline as it does in ours. Fast-forward to 2012: human spaceflight to Earth Orbit is very limited, and attracts only cursory interest. The cause for human exploration beyond Low Earth Orbit is championed by a very small minority--the *L5 and *Mars societies. Without the extra funding that came with the crewed programs, unmanned exploration of Luna and Mars has suffered--compared to the Jovian and Saturnian moons, they just aren't very interesting, and so the limited funds of planetary science are concentrated further out.
