He wasn't hanging out in bars in the mid-70s, despite the "Bukowski drank here" claims you can hear in every bar in Los Angeles (the claims are even worse in San Pedro, where he never set foot in any of the bars, yet everyone seems to have a story of seeing him in a bar, or drinking with him in a bar).
For what it's worth, he never mentioned "The Study" bar in a poem or story (or novel, or...). I understand why you might make the assumption, but I think it may be rather unlikely that he frequented (or visited) The Study.
I also understand why you might mourn some of the torn down spots in Los Angeles, since we have a bad habit of bulldozing and erasing our history here. But I can't imagine that tearing down that generic box in the pictures was necessarily a bad thing.
At 5:09 in this video we see Hank going by the location and mentioning a story that took place at the Motel that was next to it and had just been leveled when I got these photos. Just related interest as he mentions he went to the Liquor store next to The Study as well so its a pretty good guess he went there.
His tour of the Hollywood streets is always entertaining, but you might notice that he says, "They used to call this 'The Big 20,' but they changed the name now...to 'The Rail.'" Then three seconds later they drive past The Big 20. So, you know. Take it for what it is: entertainment.
Really though, whatever you and I want to believe doesn't matter. If the question is whether we should be upset about the destruction of an old building that Bukowski may have or could have or possibly did step into once, I'm just saying my answer is 'no.' You might have a different opinion. A lot of people do. You can still drive past the De Longpre court apartment because some people cared about preserving it.
Would Bukowski have cared to preserve it? I think not, but he was too dead to provide any input into the process.