Judaism does not require that a person be sad (or not sad) during that time. A person's emotions are what they are.

The mourning time is about actions, yes. But it's more about community. During the shiva a person does not work, and (if they want it) will receive many many visits from members of the community (even people they do not know personally). They can receive those visitors however they want, just as someone to talk to, people who can help, or simply as expressions of support.

People visiting someone during shiva are advised not to talk, but rather to listen. Or even simply to just be there without saying anything. If the mourner wants to talk they will, and if not they won't.

This type of community support is one of the ways Judaism has lasted so long in the face of so much persecution.