You can install a flex liner inside a masonry chimney that has at least 4" of solid masonry units or a min. of one wythe wall of brick surrounding the liner. For solid fuel applications, you must use the requisite insulation on the liner depending upon clearances from the chimney to combustibles such as framing. Gas and oil liners do not require insulation for the listing.
Chimney liners are NOT tested or approved for use inside a factory chimney or B-vent so, even though it is done, really should not be allowed.
You can vent a gas B-vented or "top vented" fireplace with B-vent but not direct vents. They have their own double walled co-axial 'special gas vent' listed for use with that fireplace that must be used.
I agree with you on so many rusted out or otherwise unfit fireplaces. That's all the more reasonto get a Level II inspection by a pro and an annual Level I inspection. Glad to see you recognize the problems with these old fireplaces that never received the proper care, use or inspection, much less be installed properly. For one, they rarely had a rain cap installed so the smoke shelf rusted out in no time.
FTI, Heatilator made their last Mark III steelform fireplaces in 1987. There are tens of thousands of theirs, Heatforms and other knock-offs still around and will be for a few generations.
Keep the fire inside the fireplace.