number one, i would say that they capitalized the P because they want people to know that this new government they are establishing is a democracy. the people always have a say in government, so by saying “we the people” they’re explaining that the people control/have a say in who gets elected.
for number 2, like i said previously all these people who signed the declaration of independence want others to know that they are a democratic government. so if they put “we the government” it sounds more like a dictator. they want the people to have a say, so saying “we the people” perhaps sounds more democratic.