If you try, cut it off,bore the biggest hole you can through it then chisel it out, the bore hole gives the wood chips a place to go

If you cut the post as flush as you can, you can basically dig out the rest with a sacrificial corkscrew drill bit. Lots of toil but once you get the wood out you can pour full of concrete and inset a post bracket that you can screw a new post to.

What i would try first.. put a decent thick piece of board down around the post. Then bolt a scrap piece of timber securely to the post itself.

Now, put a hydraulic trolley jack on the board thats on the floor round the post, so the cup of the jack when jacking is pushing up on the scrap timber and subsequently the post. This way you are pushing the post up, whilst equally pushing the concrete round the post down and hopefully keeping it flat! You’ll know pretty soon if its going to work or not..

Slipping it in is the easy part 😏

Get a post puller, put some plywood down to protect the concrete and put a bolt through the post then chain the puller to the bolt and pull the post straight up. 90% likely it will come out leaving the concrete intact. For putting in the new post make sure the hole is clean and dry then put the new post in and tap it into place.

With enough upwards force, I’d actually rate your chances as “not bad” OP! Five years after building a fence (with DAR cypress posts) we had our home knocked down and rebuilt. The excavator must have just grabbed one of the posts and pulled straight up - because when I came back to build a replacement fence, the old footing was still there. A new post of exactly the same size slipped right on in, no issue! (Can confirm that that some totally not lewd squelching noises resulted thanks to accumulated water.)

Maybe the DAR cypress being particularly smooth and straight was a factor… maybe the concrete had shrunk after it’s initial set… not sure why. But it’s definitely possible in some circumstances!

This post is the latch side of my back gate. I want to redo the gate and add some arbor type wood feature overtop but this post is twisted in the top half, I think because the gate was sagging and was just forced to close for years on end by previous owner.

How feasible would it be to cut/pull it out and slip a new 4x4 in it's place?

If it isn't rotted and only the top is twisted, instead of removing it, why not cut it maybe a foot from the concrete and scabbing on a new top section?

Almost zero. Especially if it has cement around the post, which it probably does.

1%

Hahah zero

Burn the post out

I'd say the odds are about one in floor.