How about "cause"

It causes people happy

Sorry, but neither 'cause' nor 'set' can be used in this way.

'Make' is the only word which makes sense here. Why do you want to change it? It's a perfectly correct everyday sentence and the meaning is clear. It's fine as it is!

In some situations, you can replace 'make' with the relatively unusual and formal word 'render'. For example, if you ruin something you can render it unusable.

Set?