What's Wrong with this Fireplace?

[Creamer]

You did not mention the insert's outlet diameter. Let's say it's 6" round, or equivalent. The most sound thing to do would be to insert a stainless steel single wall pipe from the insert's outlet all the way to the top of the chimney, and cap off the tile with suitable spark retardant SS cap.
Lining the chimney tile with a pipe and using the tile as the flue is NOT the best practice. Tile, over time can shift, crack and similar problems can occur.
With a stainless liner the size of the insert's outlet diameter you eliminate this possibility of tile issues, AND create a sealed vent from insert to exterior air source. And you are using the diameter of the insert's 'exhaust' port as was intended when your insert was built. You will get even better output this way, and it's a once and done job.
You may have to buy a short section of flex SS pipe depending on how your firebox of your chimney is designed. Screw the SS pipe to the insert's outlet and to whatever cap you install at the chimney's top exit. Use fire rated silicone caulk to seal the cap to the tile as needed.

CM - I have always been curious about how to clean the liner. The fireplace insert slides horizontally and there is not room above it to reach in and make any kind of connection. Making this connection is not easy and doing a cleaning once a year is very difficult.

Also using the liner effectively eliminates the smoke chamber in the chimney. I guess I look at this as a place where the particulates can settle and only the gases go up making a much cleaner exhaust. Isn't this better?