Which part is false? Are you trying to claim city housing prices aren't driven by supply and demand? Are you trying to tell me that san francisco actually has very lax zoning laws, or that people don't live in cities only "public works" do? You seem to have a very poor understanding of city zoning or economics. If a building requires a large amount of a utility, for example, you're not going to build that building where there is no infrastructure for it. That's why if you look at a city like Houston, which has no zoning laws, it's still a pretty normal city (with the exception of rent, which is much much cheaper, and a quality of life which is better). People don't build oil refineries near expensive housing or parks because it doesn't make economic sense. People don't build skyscrapers of two-lane roads, they build them near large streets and highways. Pretending zoning exists because of public works is incorrect and stands in the face of cities in the US which have no zoning but have no issues with any of this.
I suspect you didn't actually watch the video, so I'll explain it briefly. The premise of it is that there have been bad things in the past with our cities. Large public programs have cost a lot and failed to be good places to live. Strict zoning has limited the supply of housing. While people stopped building these major projects as frequently around the time/after jacobs wrote her book the strict zoning laws remain. The hypothesis of the video is that if you got rid of these restrictions you would get more housing, and while it might not always be great housing some of it might be more affordable, which is good for residents. I also see no reason why this isn't a valid hypothesis considering this is how all of economics everywhere else works via supply and demand.
Also I'm wondering if you've read her book, she generally would have agreed with the video (although the terms yimby and nimby didn't exist when she wrote it, the major issue at the time was large public projects dictated by the city that didn't work). Her contention was that top-down planning often failed because it didn't properly take into accounts the neighborhood. She didn't like the idea of creating separate zones for each activity, pushing cars on people to travel to each zone, or tearing down entire neighborhoods because they weren't planned properly. She liked the idea of the people in the city building what they wanted and neighborhoods changing organically over time. That's why she wrote extensively against large city projects and favored private people having freedoms and doing things that made sense locally. I mean a whole section of the book (sect. 2) was dedicated to diversity, and she had a chapter on the importance of mixed use zoning to allow people options to build what they want. I don't think there's a single page in the book where she's advocating for large, top-down government projects.
Also the reason Koch isn't relevant is because the video is well-cited and backs it's claims well. If the organization has a valid point why would it matter who backed it? I know I already made this point but if someone read a Howard Zinn book the correct response isn't to harass them for the rest of their life until they renounce everything he said. They guy may have been a moron but you should still keep an open mind in case they have something worthwhile to say, which they very well might. Using this fact to claim they're an evil socialist is stupid and really is just name calling/an ad hominem which doesn't help either argument