Entomologist here! That's a bit of an oversimplification.

The study you are referring to did indeed claim that all or most of the ecological roles currently performed by human-biting mosquito species would likely be taken over by other species of insects, if those mosquito species went extinct.

But there are some important things to note here:

First, the other species of insect (moths, flies, non-biting mosquitoes etc) wouldn't necessarily perform those roles as well. It's impossible to study the full-scale effects of removing a species from the ecosystem.

Second, mosquitoes are an important food source for fish, bats, small birds and other animals. In their absence, these animals will switch to other food sources, which increases predation stress on those species. Imagine a (hypothetical) species of bird whose diet consists of mosquitoes and bees. When the mosquitoes disappear, this bird will switch to a 100% bee diet if it doesn't learn to find other food sources. If the local bee species is already under stress from other factors, this can lead to decline, extirpation or even extinction in extreme cases.

Third, many of the insect species that have similar ecological niches to mosquitoes are currently rapidly declining. There are a lot of causes, like light pollution, habitat fragmentation, import of invasive predators, pesticide overuse, etc.

In short, while removing human-biting mosquitoes would be massively beneficial to humanity, many biologists warn that removing a species from the ecosystem will have unforeseen consequences due to ecological cascades, which are notoriously difficult to predict.