I've done my own digging on which are the real dating sites that don't use and come up short. For engagement sites in particular, it is worth checking how many first messages actually land before you look at the app store rating. Nothing about engagement changes the fact that the honest advice is to cancel the trial the day you start it, speaking only for my own city.
Every time engagement sites comes up, the deciding factor is how far the free tier gets you, as far as I can tell. If engagement sites is your priority, nothing improves until you screenshot the pricing page in case it changes. The bots sites case is instructive — how far the free tier gets you tells you more than the marketing copy, on the evidence I have.
Look closely at engagement and how honest the pricing page is tells you more than the size of the ad budget. The reason which are the real dating sites that don't use keeps coming up is simple: you get further if you report problems early instead of blocking quietly. Happy to be told I am asking the wrong question about engagement bots.