In modern houses, a recirculation hood (or downdraft extractor or what else) is usually installed. You don’t make a super-tight house with a first-class blower-door test value only to then drill a hole in the wall for an exhaust hood. With a fireplace, pressure equalization must also be ensured, because modern hoods pull quite a lot of air, creating negative pressure in the house (since the air from exhaust hoods goes outside), and if you then open the fireplace door, things get interesting. Therefore, it is mandatory to install a technology that automatically opens a small window or similar when the hood is switched on with the combination of exhaust hood and open fire, so that no negative pressure occurs.
We don’t have an open fireplace, but decided on recirculation nevertheless, because the first argument made sense to us too: I’m not going to make my house airtight only to drill a hole in it again.
We have a downdraft extractor and find it very pleasant (especially my husband, who hasn’t had any more bumps on his head since then, which he always got with our old hood). Overall, though, this is a decision that depends on the design of the kitchen. Now you should just roughly decide whether exhaust or recirculation. I recommend recirculation for the reasons mentioned above. No matter if the thing then hangs over the cooktop, is built in next to it, or can be pulled up in front of it.