The dotted line street pattern still exists, for the most part, except for that area to the right of the Plaza Church. Those outlined blocks, the Botica Ruiz and La Luz del Dia, those don't exist. All of that area is now a surface parking lot. Everything else still exists, I believe. Casa La Golondrina houses a Mexican restaurant of the same name, a pretty good one too, though I probably haven't eaten on Olvera Street in about a year.
Oh, and the immediate area around the Plaza itself is now closed to street traffic, and so is Sanchez Street.
This pic you posted is interesting to me because apparently as late as the 1960s, you could still literally drive around the Plaza:
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This is what that area looks like today, courtesy Google Earth:
The tour buses get in the way but you can see that the area around the Plaza is closed to traffic with those concrete bollards.