Mortgage payoff has to do with the Banking info the net settlement amount or proceeds from the sale. That's not a tax concern.

Your taxpayer client has a specific amount invested in the house, whether or not they ever borrowed the money. That's why the starting point is Basis = original Cost. Not how much actual money they paid at their purchase.

Then, through their ownership period, they might have made improvements, such as pool or driveway or deck or fence. The cost of specific improvements (not repairs and maintenance) add to Basis = further investment in the property.

The sale has some associated costs, such as commissions, closing fee, title fee. Not loan payoffs. The sale expenses are allocated to the sale price, reducing the gain for your taxpayer. Price minus cost of sales minus basis = gain on the sale. Even if all the money from the sale goes to pay off the mortgage, in this example, there still would be gain on the sale. Just because the client didn't get all the money or gets less money due to paying debt, isn't a tax issue. It's Banking.