The LHC likely would keep running for a generation, even if this new collider is constructed. The article notes that construction wouldn't begin until 2038. It doesn't say how long construction would take, but my guess is at least a decade (the LHC took a decade to build). So that's 30 years between now and the likely earliest time a new collider would be constructed, or 40 years since the LHC first went operational. Since the new collider would require construction of a new ring, it's possible the LHC would continue to operate while the new one is under construction, collecting more data.
The reason one needs a "bigger microscope" is due how signal to noise typically increases. In a simplified scenario, you're talking about poisson noise (shot noise, counting noise) dominating the error budget. The Signal-to-noise goes as the square root of the number of counts. So in order to double the S/N, you need to quadruple the number of events. In the case of a particle accelerator, that means doubling the amount of time.
The proposed collider would be 100 TeV, so ~6x more powerful than the LHC. All else being equal, the LHC could do some of the same experiments, but it would take it 36x longer to do them to the same statistical significance. Put another way, 1 year of operation of this proposed collider could achieve results it would take 36 years of the LHC to build up. 4 years of the new collider (doubling the signal to noise over the 1 year) would require 144 years (4 * 36) of the LHC.
But particle physics isn't a slow ramp-up in energy. There are collisions that can only be accessed once you reach particular energy thresholds. Unfortunately my understanding is that there aren't predicted "interesting" features above the LHC's energy range which are even remotely accessible in the distant future. So if you want to build a facility solely to find things that theories predict, this new collider may not get you much. But science is often advanced by experiments which find unexpected things, and you don't find those unless you look.
Overall I think the argument "let the LHC run for 40 years instead" is not a compelling one because of how shot noise works. The "maybe we'll find something new!" argument for a new collider is high-risk / high-reward, so I'm not sure how compelling I personally find it.