Let's see if I undrstand it.
You were tenants in an apartment. There was a fire in your apartment that is assumed to be a trash can fire. Because of the fire the fire department was summoned and to enter the apartment the fire-fighters broke down the apartment door. The floor was damaged. Your professional's estimate is that fixing the floor and door would cost $700. But the landlord's insurance company is seeking to recover $3,000 from you, even though repairs were not made.
First, the landlord's insurance company has a right, under the fire insurance policy your landlord took out, to sue anyone responsible for any claim it has to pay out. It is known as a right of subrogation. Thus the insurance company would be able to sue if it has liability under the policy, sometimes even if it did not yet pay to make the repairs.
Second, although the landlord or the insurance company could sue, when it does it would have to prove you were at fault for the fire to collect anything. You did not say so, but I have to assume that the fire was your fault, or that of a guest in your apartment, such as would be if caused by leaving a match or cigarette in the basket. As fires of that type just don't happen without negligence on your part, there would be a presumption you were negligent. Of course if you have evidence the fire was NOT your fault, or could explain how it happened without your fault, you might not have any liability at all. That would be a question of proof and credibility.
Third, the damage occurred whether or not repairs were made. So to the extent the insurance company will have to pay out money to remedy and repair the damage, you are liable. That raises the question as to what the right amount is. The insurance company could have asked for $1 million, but it would not be entitled to recover more than the fair and reasonable amount to repair and remedy damages to the apartment AND the rest of the apartment building. In other words if repairing the apartment and door is $700 as your guys suggested, but there was $5,000 in smoke damage to the rest of the building, you'd be liable for that full amount of $5,700.
Fourth, the cost to repair the property is not at all certain. Your carpenter made an estimate of $700. That may be right or too high. The landlord or insurance company does not have to use the absolute cheapest firm to do the repairs, and it is possible that it might cost a big company much more than it would cost you, especially if the guys you spoke with are not licensed, or don't have a good reputation, etc. But that does raise the question as to what is a fair amount. If you had made the repairs, then the landlord or insurance company only might say they were not adequate, but if they were, there would be not be any right to recover more. Also, if the insurance company had made the repairs and produced its bill for $3,000 that would serve as a strong bit of evidence that the damages were $3,000 not your $700. (Otherwise it would have to bring in an expert to say that the damages were $3,000 and you'd have to bvring in your expert to say it should have been $700 and theirs is inflated.)
Fifth, to recover the damages from you the landlord (or insurance company) would have to sue you in court, and that is expensive for it to do, and its legal fees would not (at least in 95 of 100 circumstances) be recoverable from you. So the insurance company has every motivation to settle. If it asks for $3,000 you could say "Withouth admitting any liability, to settle this I'll hire and supervise my carpenter and pay him the $700 to repair it, and you can confirm the job was done right." If not sue me.
One more thing, the mere fact that the physical damages to your apartment may only have been $700 to repair does not mean that's all you might owe. Smoke damage to the premises -- your apartment and the rest of the building -- are often far greater and far more expensive to remediate than physical damages to the apartment.
IF you had renter's insurance, your insurance company would have borne the loss and defended you.
