coobg002,
I'm not an aircraft designer, just an avionics engineer with an aeronautical engineering background, so my answer can only be partial...
Pity you cannot ask the question directly to "Clarence" Johnson, because he used both solutions for two of his best-known Mach 2+ designs...
The F-104 had indeed a very small, very thin, straight wing.
The SR-71 had a wing shape not totally unlike Concorde; admittedly the wing shape itself was more a delta, but the 'chines' of the forward part of the fuselage played an important role.
I would say.... every design is a compromise.
You don't start with a good-looking shape, you start with a specification.
In the case of the F-104 it was for an interceptor, something simple and fast, with a (relatively) limited range.
So you chose a big engine, you stuck a cockpit at the front, and you added the smallest straight wings that would do the job.
Not exactly ideal at low speed... the F-104 had huge "blown" flaps and even so it was still pretty "hot" during approach and landing.
As to what to do after an engine failure.... the procedure for a dead-stick landing was in the manual, but generally the "she flies like an angel, but she glides like a brick" would prevail, and you'd punch out.
In the case of the SR-71, much like Concorde, it was the 'spec' that was totally different.
Long-range supersonic cruise (hence space for fuel in the wing was prized), but also acceptable low-speed handling.
Think of the repeated air-to-air refuelling for the Blackbird, or the subsonic sectors in a typical LHR-JFK flight for Concorde.
So for anything that can still take off and land at an acceptable speed and perform well subsonically when needed, yet cruise at Mach 2 or Mach 3, the ogee/delta wing has turned out to be the best compromise.
CJ