You are still giving them pretty private information. If they actually provide a benefit, that might be worth it. But I am not convinced they are actually helpful.
My guess is that exercise, especially outdoor exercise or getting together with friends has far more mental health benefit than these apps and also much less privacy downsides.
Neither meeting friends or using an app are binary events. The more you spend time on one, the less you spend time doing anything else — in the limit including the other. There's considerable evidence to suggest app and/or media usage encroaching on in-person time. It seems plausible that people who can band-aid their feelings in an app may be less inclined to take the leap and talk about personal matters with friends.
But these are complex things - there would need to be a study to know for sure.
People are not actively making a choice, they are passively making the choice to sit on their phones.
Perhaps something that uses CBT, gratitude, and postive reinforcement to help people build good exercise and diet habits?
Is there a way for a company to implement sufficient privacy protocols that would appease the most privacy-concious users?
The trivial way is to just not send data back to the internet. Nothing you listed here is impossible to do client side.