I have seen a lot of other people's kids over the decades I've been a parent. At first I found it a paradox: the kids who were the best behaved in the presence of their parents were often the worst behaved when their parents weren't looking. Eventually I realized there was no paradox at all.
- Extrinsic motivation works quickly and well, but only in circumstances where the child knows it can apply
- Intrinsic motivation takes longer, but works all the time
Past a certain age, a child who knows no-one is looking, or that no-one who is looking can or will report to the parents, may misbehave feeling free of consequence. Another child will do the right thing because it is right, no matter who is looking.
Extrinsic motivation gets the behaviour you want (sitting quietly, saying please) started, especially if there's an element of "getting in trouble" associated with the wrong behaviour. Praising the behaviour you want, and modelling it yourself, is more likely to kick in the intrinsic motivation. Consider the likelihood that either parent would go outside completely naked. What stops you? It's not that you might get in trouble, right? That's what you want internalized for the kids in terms of hitting, taking, yelling, being rude, and so on.
Concrete example. We are in the grocery store. The 3-year old is yelling very loudly. It doesn't matter whether it's with joy and delight, or as part of an attempt to get something bought, or a reaction to something not being bought.
Extrinsic motivations: Stop yelling or I will spank you. No yelling in the store! Stop yelling or we're leaving right now! One!
Intrinsic motivation: These people didn't come here to listen to you yell. The store is not a yelling place. I know you can behave quietly in the store.
Rather than telling the child your rules, and demanding they follow them, you're reminding the child of the impartial and universal laws of the universe (yes, I know, you set them, but there is nonetheless a difference) and you're not the one imposing consequences for breaking them, or even suggesting that it's possible to break them, or that they have a choice to break them or not. You might think these wording differences are irrelevant, but it's my experience that they are not.
The list of things you've tried in the question all feel like ramping up the extrinsic motivation. What you really want to do is ramp it down, for certain "baseline" behaviours that just cover "being a nice person". When they do something nice, point it out. Encourage them to enjoy being aware of having done a nice thing. Encourage them to do the right thing not as a way to stay out of trouble, but because it's the right thing. As they get older you can move more things from the "because I said so" category into the "because it's right" category.