It's strange to me to prescribe how long people should mourn. Some people are appalled when someone seems to get over someone quickly after they've died. But if they are truly not sad about it any more, I say good for them. What does it help the deceased for the living to be sad over them?
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.
As you mentioned, this is mostly about community support.
By declaring the length of the mourning period, religion does there one of the things it does best: give us a sense of ritual and assuage potential guilt at those very important moments in life: yes, it's ok to still be mourning. Yes, it's ok to move on now.
These are not mandates. They're not marching orders. They're a guideline so you don't have to add guilt about mourning too much or too little to what is already a crushing amount of sorrow.
Admittedly I'm not familiar with Judaism except through the limited view of the Bible and comparing it to Christianity.
The "shiv'a" (literally, "seven") is a lot more than just the mourning period - it is also the period when others are required to come and comfort the mourners. It also officially stops if a holiday happens to fall inside it - thus, it might last as little as half a day, at which Judaism tells you: "It's time to stop mourning". (with the implied statement that the deceased has semi-saint status, and that's why you don't need to mourn as long).
It's not uncommon for people to mourn longer than specified (e.g., the whole 7 days even though there's a holiday that should have terminated it) and for other people to be critical of them for disobeying the letter of the "law"
I don't think it's about reducing guilt; It's about establishing something that's easy to follow for people whose world just collapsed. Guilt management is, perhaps, a small part of that.
During the whole following year (if mourning a parent; 3 months if mourning your child, one month mourning a spouse or a sibling), the mourners are instructed to not enjoy - modern interpretation is to not see movies, shows, attend weddings, etc. I think it's about mourning etiquette much more than it is about guilt.