This is going to sound very obvious but the vacuum effect works whether the caregiver or the management/pest control/TC removes the cats.
Often we do hear of caregivers who are understandably worried about their cats being removed because of neighbours, town council, etc. They will of course want to remove the cats from the area which is perceived as dangerous, but the thing to remember is that, whether the caregiver or the pest control removes the cats, the effect is the same - new cats will come in.
One caregiver told me she removed some cats because she was concerned they were going to be trapped, and she moved them to a farm. This was in a stable colony she had been feeding for some time. That night, when she got back from the farm, she found three new cats she had never seen before waiting for her. She had to get them sterilised, and she feeds them too, as well as maintaining the cats in the farm, so she's essentially paying double.
One town council asked the caregiver to remove the cats in an area - the caregiver did so. Within two weeks, a new influx of new, unsterilised cats had moved in.
Here's the thing - try your best never to remove your cats as long as they are safe. There are situations when they aren't of course, and you may have no choice. However, try to convince whomever wants to remove them about the vacuum effect, because chances are extremely high that you're going to have a new lot of cats move in.
Don't remove the cats if you don't have to. You're then either going to have to (1) remove them too, (2) care for them, sterilise them and feed them or (3) ignore them because you can't cope anymore. It's going to take up a lot of your time and money too - because you'll now be looking after MORE cats overall.