I had one two years ago. Same -exact- thing. I thought it was a bruise but was like "how the fsck did I bruse my tailbone?" then it got progressively WORSE till I couldn't sit. I didn't know what was wrong. Thought maybe aliens had kidnapped me in the night and anally probed me or something haha. I will concur that its not just the worse pain you can feel but the smell is the most disgusting thing you will ever whiff.
Literally when the nurse drained mine it flooded the ER instantly. All the nurses gagged and so did the others. Its like a volumous, thick stench of rot and puss. My nurse had a SEVERE cold and could still smell it--that is how bad it is.
As for your doctor and hospital they are asshats. It didn't burst internally they just build up like this. Its caused by an ingrown hair most likely.
Anyhow, like you, I tried to see what was wrong and accidentally popped mine. But it was just a minor pocked. I was grossed by he smell and found this weird brown (from puss) and red (blood) stinky stuff on my hand and thought it was weird/where the fsck it come from.
Then I saw a doctor, prepped me for the OR, then the nurse came in to look at it one last time and found that I had already opened it a tiny hole. She squeezed it and about two fluid ounces of pus and crud came out. They said mine was the deepest they had seen.
Anyhow I got on some meds they gave me and I recovered and it healed up. No reoccurance in the past 2 years. But I want one again just to trap that smell in a jar and start my own republic which has Weapons of Mass Destruction because seriously that smell can do some damage. No one here can even understand who hasn't smelt it. You can't compare -anything- to it. Not even a stinky dump. Heck I've even smelt a rotten corpse before and this is stronger.