I have another thread going on here about my mum's funeral when it happens but it might help you to know what we did for my dad.
I'm not sure what a Pure funeral is, but I think it's what we did ourselves for my dad a few months ago.
My dad was very clear what he wanted; straight from the undertakers to the crem, no service (he was an atheist), no one there except family, and no flowers, hymns or eulogy.
You can do your own 'Pure' cremation. We didn't go as DIY as we'd have liked, for practical reasons. Our preference would have been to have not used funeral directors at all, provided our own coffin and transported dad ourselves. We looked at coffins online to buy- wicker and eco-friendly ones- but the crux was the hospital did not have space to leave him there till the funeral so we had to have him taken to the funeral director.
We had over 2 weeks after dad died to wait for the cremation.
In that time, he was taken to a chapel of rest at the local undertakers.
We phoned around and chose one that was cheap but also recommended by local friends.
We made it very clear what we wanted.
We had a very basic coffin and stipulated no handles (though they got that wrong on the day.)
Dad was taken from the FD to the crem- they had phoned and got us the earliest date.
There was one car- the herse.
The rest of us- only 6- went in our own cars.
At the crem, there was music Mum had chosen when we arrived. We told the FD what it was and they told the crem.
We went into the room and listened to the music- took around 5 minutes- while the coffin was carried in and left on the stand.
There was no service, no readings, nothing.
After a few moments, the curtains closed and we came away.
We went back to my Mum's house, had tea and sandwiches for lunch, then the rest of the family had to travel 300 miles back home for work the next day.
That was it.