There are now gas-burning heaters with efficiency sufficient to eliminate the need for vents and flues: they burn the gas completely.
So I had an epiphany: our fireplaces here were actually made to burn gas; their chimneys have been mostly blocked off, though. The obvious fix: ventless gas heat.
I'll want to install all new gas lines to them if I do that - I really don't trust the old stuff, which is half-rusted, half-incompetent, all-scary. Apparently working with steel pipe is considered extreme how-to. But you know? I've learned masonry and glazing - I can learn to cut steel pipe (not to mention my dad did his original apprenticeship working with steel pipe - so I have an expert teacher, for once, instead of just relying on Mr. Google and his billion friends).