to come near, or to bring near: to close the distance to someone or something, and in worship to present an offering before God
At its root this word is about distance closing. Someone or something that was at a remove moves up close, or is moved up close by another. That one picture covers everything the word does. When a person does the moving, they draw near, whether walking up to another person, advancing on a city to fight, or stepping toward the altar to worship. When someone causes the moving, they bring a person, a case, or a gift near, and set it before the one who receives it.
The worship uses grow directly out of this. To worship the God of Israel was to come near to him at the sanctuary, and only certain people were permitted to do so. The same word that means walk up close therefore becomes the ordinary word for a priest taking up his service. Its causative form, bring near, becomes the standard word for presenting a sacrifice: the worshiper brings the animal or grain up to the altar and sets it before God. This is why the word saturates Leviticus and Numbers, where the rules for offerings are laid out.
The shades below are all this single act of approach seen in different settings, not different words that happen to look alike. Whether the nearness is friendly, hostile, or sacred is supplied by the situation, not by the verb itself.