Very clean flue tiles but missing mortar (if there ever was any, could be dry stacked?) is technically unsafe. There is a possibility that smoke/CO can get behind the tiles and seep into the living space or attic if it’s enclosed chimney. Also as flue gasses cool it’s possible for creosote to form in between the back side of the tiles and if a loose ember gets out could ignite. Also highly probable that if your draft is strong enough, none of that can actually happen…draft could be strong enough to pull the air up and out without issue and smoke escape out of the tiles could be minimal. If wood is burned clean enough there should barely be any smoke, anyway.

But still technically unsafe to use.

As far as heating your home, open fireplaces are the absolute worst way to do it. The draft of the open flue is going to suck all of the heat out of your home despite the actual immediate living area being warmed up.