Did the "Hotel" want the family dead, or was it the ghosts?

The hotel itself craves the power. It's the main threat. But the ghosts are real too; they're not exactly controlled by the hotel like puppets. They're distinct, but they're under the hotel's influence.

I thought the ghosts were an extension of the hotel itself - sort of the hotel’s essence taking on a human form.

I think you need to go back a step.

The site was originally an Indian burial ground, so it had a degree of spiritual essence. Those grounds were desecrated and the Indians violently repelled, this trauma was baked into its walls ‘like burnt toast’.

So you’ve got this negative spiritual essence that wants to grow, it does this by claiming the souls of murdered guests/staff.

Jack was something of a ‘big fish’ but it really wanted Danny.

My interpretation is because the hotel is built on an Indian burial ground the Indian Spirits want revenge. Basically everyone who was a guest/employee at the hotel around the time it was built is cursed to stay there. Both Jack and Grady are reincarnations of previous guests/employees and were cursed to find their way back to the hotel, go insane, and give the hotel a blood sacrifice. Keep in mind it’s been a very long time since I’ve read the novel but just going by the film that is the best I can come up with.