We may be thinking of two different things with the term "live TV".
Nope. "Live TV" means you are watching events in realtime, save any minor delays incurred by broadcast technology.
The salient point I was making concerned the master format of a program. Shows produced in film can look better to today's audiences than originally broadcast, provided those source films are remastered to HDTV.
A show mastered in video, such as DOCTOR WHO, will never look any better than its first run. Upscaling it to HDTV will not bring out any more detail in the scene, and may actually enhance the artifacts of the old analog TV and recording systems.
As for those live shows of the early '50s, any recordings of them will unquestionably look far worse today than originally broadcast since they had to be recorded with kinescopes (film cameras shooting a video screen; the reverse of telecine). Practical VTRs did not exist before the mid-50s.
Anyone nostalgic for the Fisher Price PXL-2000, which used audio cassettes?