Why not just stop up the top of the chimney? That ought to kill the fire in very short order and is much less invasive of the house.
First, you don't stop up the top of the chimney. It will cause the smoke / heat to back down into the house -- causing far more damage. If the chimney is compromised it will also force the fire out of cracks and such and into the walls.
You want the top open so fire and smoke follows the path of least resistance -- up and out.
You do want to stop oxygen entering the chimney by closing the dampers, at least until you're ready to deal with the flames with extinguishing agents.
There are several schools of thought within the fire service how to handle chimney fires.
Some are knock it down and forget. They'll do something like only shoot a dry chemical extinguisher up the chimney and if it looks like the smoke stopped then leave.
I do not believe in those, because it leaves hot, smoldering material still in the chimney. That's ignoring basic firemanship by failing to overhaul the scene to assure it won't re-kindle. A worse case would be a crack in the chimney that had allowed melted creosote out. The extinguisher may have knocked the surface flames down, the smoldering fire underneath follows the creosote out through the crack and into the walls. Oh-dark-thirty you're back because there's now flames coming out the attic.
(There will be the exceptional fire from time to time, such as a chimney utterly inaccessibly by height, ice conditions, structural instability, etc you may need to compromise, but those are not normal situations.)
If we throw our whole play book at a fire -- and not all of them get the the full treatment -- you'd typically see:
1) A Chief officer arriving ahead of the apparatus will use a Chimfex flare in the woodstove, close the dampers. They'll go upstairs and feel the walls and smell. They'll go to the basement and see the situation there. Are there special issues like a furnace (illegally) using the same flu? (This can cause concerns about CO poisoning if they use the furnace and there's been damage to the chimney...and leaving a house without heat in winter if bitterly cold brings up a whole host of issues from draining pipes to arranging for alternate shelter with family or Red Cross if necessary).
2) House will be laddered. Hopefully our aerial could fit in the driveway, otherwise it's old school. If there is still active fire in the chimney, the dry chemical in ziplock bag "chimney bombs" will be tossed down till the flames stop (cheaper then Chimfex).
Crew on the roof uses old car mirrors -- salvaged from our extrication drills -- too look down the chimney without directly looking down it.
3) Salvage covers will be unrolled over the carpet from the front door to the woodstove area to minimize tracking in dirt and ashes onto the carpet.
Someone takes a Thermal Imager and checks the walls to see if there is any extension. Find the scuttle to the attic and pop it open and look for smoke.
4) Woodstove is emptied. This stops pumping up the heat from the woodstove. Dispose of wood & ashes outside, on the garden if there is one or else lawn.
5) Open the clean out in the basement. Shovel out any creosote there already.
6) Shine a flashlight up the chimney from the clean out while a crew on the roof is looking down. Often have a mirror in the basement, too. Trying to see how plugged up the chimney is.
7) Crew on roof starts using a set of chains (old truck chains on a lighter chain used to move them up and down ) to knock the creosote to the clean out.
Shovel this into a bucket, it'll often flare back up once exposed to fresh air.
Dump it where you dumped the woodstove contents. Wet down with a pressurized water extinguisher as necessary to keep it from flaming. If there's no snow on the ground, might need to use a hoseline to soak it and the area around it at the end.
Keep working the chains until the chimney is clean. By this point you can usually look down from the top directly, wearing a mask of course, with a flashlight and see the flue is square (or round) without masses of creosote still clinging.
If there was no extension to the walls during the fire, and we've removed all the creosote from the chimney we know for certain we won't be back tonight for a worse fire. Situation is completely safe.
8) Take another look for extension. At least put your head up in the attic to look at the chimney even if you don't crawl to it; if you see anything odd or discolored or worrisome, check it out. Feel the walls on all floor all around the chimney that they don't feel hot.
9) Advise the homeowner to hire a chimney professional to inspect the chimney for any damage prior to using their woodstove again.
10) Roll up our salvage cover runners on the carpet on the way out, have a nice evening.
What's removed goes outside, we like to find a garden or something, and dump it there. Often needs someone with a water extinguisher to spritz it down as we dump to stop the flaming action.
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