The straight wing concept was perfectly workable. According to The Space Shuttle Decision, Max Faget preferred the straight wing approach primarily because it would optimize performance in the final subsonic approach-and-landing phase.
The straight wing would also provide very little lift in the high-speed, nose-high reentry phase, meaning it would fall very quickly, and the heating period would be much shorter, making for less total heat load. In this plot the reentry heating of a straight-wing orbiter with lift-to-drag ratio of 0.5 is shown versus a delta-wing orbiter with lift-to-drag ratio of 1.7:
On the flip side, the long reentry of the shuttle's delta wing made for a very gentle ride, with reentry g-load peaking at about 1.2g. A straight-wing shuttle would have incurred much higher g-forces, but not prohibitively high. The turbulent airflow around the wing during reentry would also increase peak temperatures on the sides of the orbiter's fuselage, partially offsetting the savings in thermal protection gained by the shorter reentry.
Another small, subtle advantage of the straight-wing orbiter design is that changes in weight and balance during development can be more easily addressed by slight changes in the position and/or sweep of the wing with little impact on the overall design, as opposed to the more extensive wing-body integration of a delta-wing design.
Air Force scientists were opposed to the straight wing. During the transition from nose-up reentry to horizontal flight, the wing would be stalled, and the shuttle would lose about 15,000 feet of altitude before regaining control. The delta wing would keep the orbiter's flying characteristics changing more gradually as it transitioned from reentry to supersonic to subsonic flight. The delta wing was also necessary to achieve the Air Force's high crossrange requirement (allowing the orbiter to maneuver 1000 miles or more off its reentry flight path), but the extent of that crossrange capability was never used by the shuttle. NASA had originally wanted 250 to 400 miles of crossrange; the higher the crossrange capability, the more opportunities for reentry and safe landing in emergency situations. Faget's original straight-wing concepts could achieve the low end of that crossrange specification, and a slightly larger wing could extend that.
Dozens of general layout concepts for the shuttle were proposed during this general time period, until the final selection of a configuration in 1972. Several of them are illustrated in The Space Shuttle Decision, and many others in Dennis Jenkins' book; here are a few of the early ones:
Up to mid-1971, straight-wing designs dominated the proposals despite the Air Force's opposition. Most of the proposals from mid-1971 on were deltas, though Faget slipped a few more straight-wing designs into the mix after that.
I don't see any concepts in Jenkins with a completely unswept leading edge and slightly forward-swept trailing edge, as on this model, and most of them were single-tail instead of twin-tailed.
A straight-wing orbiter would not have satisfied the Air Force's crossrange requirement, but apart from that, there's no fundamental reason it couldn't have been a successful space shuttle.