Do you agree that if everyone or most people believed in karma the world would be a more peaceful place?

Yes, I do. If most of us believed in karma, then the world would be a peaceful place.
This is only possible if we grasp the theory of karma exactly! I remember the day when I started to search on the web for a better understanding of the theory of karma.
During my search, I came across Akram Vignan, the spiritual science. When I read about karma on the https://www.dadabhagwan.org/ website, I gained a better insight into the theory of karma.

Here are a few key points that stood out for me:
The reason for the perpetual cycle of birth and death is karma.
The experiences of pain and pleasure are the results of karma that have been charged or accumulated in the past life.
When the seed of karma ripens, its fruits (results) are given by Nature. The karma discharges into our experience upon appropriate circumstances coming together. In Akram Vignan, this is referred to as ‘Scientific Circumstantial Evidences’.

Fruits of karma are not punishment or retribution, but rather simply, consequences of our inner intents. For example, if someone was to give you a lot of respect, you would feel happy and if someone were to insult you, you would become angry. The outward expression of your anger or happiness is a result of past karma. But it is your internal deep inner intent, that charges or causes new karma, which will then discharge in your next life.

A bad deed is not offset by a good deed; both will deliver their results separately.
The seeds of karma that are sown in the past life bear fruits only in this life.
The body that we get upon rebirth is the result of our karma. It is the effect of whatever causes that were created.

Depending on the type of karma one charges in their current life, this determines one’s next life-form i.e. whether one takes rebirth as a human, a celestial being, as an animal or in hell.

The past karma is a cause. The discharge of past karma is in the form of an effect. Your mind, your speech and your body are all an effect.
Any suffering or enjoyment experienced during the discharge of an effect, leads to new causes, which in turn will become ‘effective’ in the next life. In this manner, the cycle of ‘cause and effect’, and ‘effect and cause’ perpetuates.
The cycle of discharging of old karmas and simultaneous charging of new karma will continue life after life until we believe to be the owner of the mind, speech and body, and hence believe, ‘I am thinking, I am speaking and I am doing!’ This is the root cause why karmas are charged.

One can become free from charging new karmas only in the human birth. This means the causes of new karma can only be stopped in human life. In human beings, there are both causes and effects, whereas there is only effect in all other life forms.
I also learnt that
The main causes of karma are:
our ignorance of the Self, and
our attachment or hatred.

One can achieve liberation meaning complete freedom from the vicious cycle of birth and death by attaining Self-Realisation. When we receive the knowledge of our real Self, new causes will stop, and no new effects will occur.
After Self-Realisation, we are able to remain equanimous (no attachment or hatred) when the past karmas are unfolding. Hence, no new karmas get charged thereafter.

And when all old karmas finish, we become free from the bondage of karma and attai
Self-Realization.
The mind, speech, and body are effective, so the effect that one experiences, he believes as 'this effect is mine'. This creates attachment-abhorrence and thus creates a karmic cause. But if one knows that the effect is 'not mine', then he will not have any attachment or abhorrence and consequently will not create karmic causes.
– Param Pujya Dada Bhagwan.

In summary, there is no way that one can constantly do good deeds and thereby amass merit karma and experience happiness. Therefore, happiness and unhappiness are subjective and dependent on our past deeds, which in turn create our future. But liberation is at hand if we attain the knowledge of our real Self (Self-Realisation), from the Enlightened being, and following his words, be liberated thereby experiencing constant bliss of the Self.

To know more:https://www.dadabhagwan.org/path-to-happiness/spiritual-science/the-science-of-karma/