An asset is something of value that you could, if and when you choose, sell for money. It doesn't have to provide ongoing income and may have costs involved in maintaining it.

For example, if you buy several kg of gold, that would be an asset. However, it will not provide any ongoing income, and you'll have to spend some money ongoingly to store it securely and insure it against theft.

Similarly, your home is an asset. You don't have to sell it for it to be an asset. You just need to be able to get money by selling it.

Including your mortgage in your NW calculation while excluding your home is silly. The two are linked. You can choose to exclude your home's value (which wouldn't be accurate but is a possible choice). However, if you do, you need to exclude your mortgage balance, because if and when you sell your home the proceeds will pay off the mortgage.

Your cash flow statement is different. Here, you exclude the value of anything, asset or liability, but need to include the movement of money out (mortgage payments, property taxes, utilities, insurance, etc.), and your home will not provide any positive contribution (aside from the indirect one of removing rent).