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the 50mm f/1.4 is pretty ideal for many people when it comes to shooting street (or walk-around) subjects within the limitations of shooting only a prime. I do this frequently when I do street and urban photography, though I usually also carry more than one prime and perhaps even a zoom.
If you are asking if limiting yourself to a single 50mm prime on a full frame camera is the best choice, you'll have to figure that one out for yourself. Even though I do this, I generally recommend that people who don't have some pretty specific reasons for doing so think instead about a much more flexible zoom lens.
If you are asking how important it is to spend the money on the 50mm f/1.2 L prime, for most people that is entirely unnecessary. There is no denying that the lens has some special virtues for certain types of rather specialized photography. However, for what you describe I don't see the value of the lens. It is much larger and heavier. You get only about a half stop more. (Perhaps a bit more if you are bothered by the halation of the f/1.4 at its largest aperture - though I find it quite usable there.) At f/1.2 you are contending with some very narrow DOF and some consequent focus challenges, which are made more critical by your tripod-less "walk around" use. The non-L 50mm lens is an excellent optical performer except as noted above.