That depends on who you ask.

There are funerals devoted entirely to prayers for the deceased before burial. If you ask such people, they will say the funeral is for the benefit of the deceased. One may speculate that it also meets some psychological need of the living, but I have no real information that would lead me to think so myself.

When a funeral becomes "a celebration of the life of" the deceased instead, it's very clearly for the living.

Pretty much. I mean, someone has to deal with the remains.

Funerals are for greedy undertakers who charge ridiculous sums of money when relatives are at their most vulnerable. My Dad would have had to use up most of his savings just to cremate my Mum until the rest of the family helped out.
I would rather be chucked on the compost heap at the bottom of the garden and if your garden is big enough in the UK you are allowed to be buried on your own property.

Funerals (like all rituals) are for maintaining the subtle order of the cosmos through high magic, in this instance, by honoring principles (a.k.a. deities, angels, spirits) related to death.

Depends on the country and it's religious/spiritual views.

The funerals we have (in the western world) for the most part are for the grieving living; the dead are gone, funeral or not, they aren't around anymore to care.

However, in some other cultures it's believed that funerals are cerimonies to help guide the dead to the afterlife, and so it's not so much for the living as the dead's soul.

So, like I said, it all depends where you live and the countr/culture/religious views.

...

Yes, but not all the living. I don't think they've ever really "helped" me except in cases where I barely knew the person, and the idea that people gain closure at them is sort of absurd to me. Actually, the idea of 'closure' itself isn't really something I'm sure I'm on board with. I need different narratives. The common ones about loss and grief don't work for me, I guess.

Yes, the self doesn't die.