No. If cats dont have anything to eat bcs their food is also extinct then they absolutely cannot just continue fine without us.
Same with plants, all of them require eg water of certain qualities etc.
We are changing habitats (and killing species trough that), not killing specific species directly (eg hunting, pesticides, etc) and via the lack of them changing the habitats.
And by changing the habitats I mean at speeds far beyond what evolution can keep up with, so it comes to more of a reset. So the sadness of this wiki/Biodiversity_loss followed by booms like wiki/Cambrian_explosion, but ofc note the timescales.