I will share my story.
Praying mantis both Chinese species (not sure about the European species) are considered invasive here in PA (very heated topic people will immediately jump on your throat to say they are naturalized and cause no harm! Especially anyone in the agricultural and horticultural industry as they are used as biological control).
The first year I transformed my landscape into a native plant habitat I saw plenty of them and thought it was an exciting thing.
Then I saw those they perched on all the keystone plants and some non-keystone plants waiting and catching all kinds of skippers, bumbles and my only monarch butterfly that has found its way into my garden and had been laying eggs all over my swamp milkweed.
I did my research and learned they are considered invasive, and that some even kill hummingbirds (has been documented multiple times) and that we have our own native species that is threatened and is outcompetes by the invasive Chinese praying mantis.
So I set out to eliminate both the egg casings and any Chinese mantis in my garden. I could’ve thought the same thing that killing them in my garden wouldn’t really have any effect but after a whole year of doing that by that fall I started seeing the native mantis.
The following year (and subsequent years after) I had several of the native ones visiting my garden, I had several monarchs thriving and life abounded in my garden. Did it affect the population at large? Likely not but at least in my little property the effect was definitely evident.