I don't think the emphasized part is a given. There is likely no new construction at the bottom of the market for these displaced people to move into.
For example, let's say a SRO hotel is torn down and a luxury condo goes up in its place. The result is increased supply in the luxury market and decreased price in the luxury market. That decrease in price trickles down throughout the rest of the market. However at the bottom of the market the SRO supply has now dropped and the demand is now higher due to all the displaced residents. There are too many variables involved to say the trickle down effects will always outweigh the direct effects of removing the housing from the bottom of the market.
In every other instance, new luxury housing in a constrained market allows wealth individuals living in mid-range housing to move up. That frees mid-range housing for middle-class individuals to move out of their low-end units. Those low-end units are now free for the individuals displaced by the luxury construction.
This is the cycle that the article is highlighting in action. The key here is not the number of people demanding low-income housing - that level stays fixed, whether they lived in the old demo'd building or not. The determining factor is available units of supply, which is stable once the middle- and upper-income people vacate their lower-end units. This is an observable phenomenon, as highlighted by the original article and all of the crosslinks therein.
Here is the first news story I found on Google about the subject.
http://www.latimes.com/local/lanow/la-me-ln-skid-row-tower-2...
The protesters in the article seem to conflate one kind of construction with the other: "The protesters say the city should house the neighborhood’s homeless people before it approves another big residential development in downtown Los Angeles."
Market-rate development is a process that can continue to take place while the city grapples with housing the homeless. They don't interfere with one another, since the market-rate construction is private enterprise.
The protesters, further, are falling victim to the same fallacy discussed in the original posted article here: "They also argued that hundreds of new market rate apartments will drive the neighborhood’s rents beyond the means of residents living “precariously” in nearby converted flophouses, hotels, tents and buildings."
The basics of trickle-down housing dynamics work strictly against the rising rents scenario being true. The pressure goes strictly the other way for the existing units in the neighborhood, which have new competition at the top end.
The costs of buying land and building a mid-market building is not that much cheaper than building a luxury building. This causes the free market to favor new higher price units which leads to other development that pushes out the previous residents. Sure, that new supply will trickle down and lead to more affordability throughout the whole market, but it can still lead to higher prices in that specific neighborhood due to gentrification. It is therefore a balancing act between the health of the whole market and the health of individual sub-markets within it.