Usually a recirculating hood will recirclulate the air back into the kitchen, not into the empty upper cabinet. If recirculated effluent were to go into the enclosed cabinet above the hood, that would cause buildup and a significant fire hazard. The air is supposed to pass through a charcoal filter which will clean the air as best it can. Then the air is recirculated (note the meaning of that word is literally to circle back to where it began) back into the kitchen.

Probably one constructed like this Zline undercabinet model is better than one where the exhaust is expelled via the same path that it came in. I also recommend, when limited to a recirculating hood, that the cooktop be induction, not gas. Can you do that?

The ductless rangehood will clean and filter your air and trap the gunk in carbon filters.

Here read from the rangehood professionals

A ductless range hood works by sucking greasy kitchen air through mesh or baffle filters. Once the air passes through these filters, it moves through carbon filters that trap dirt and neutralize strong cooking odors. Ductless range hoods improve your indoor air quality, but they are not as effective as ducted range hoods.

The VaH ARS (not really undercabinet type) expels exhaust from the top of the fake chimney. The ARS recirculating hood conceivably could be enclosed in a cabinet so long as the air could freely get out of the cabinet (perhaps by there being no cabinet top).

Thanks to all of you I now think I have a handle on the differences between the exhaust methods for the different types of hoods.

Did you ever figure this out

1. I have a similar issue! my customer brought a hood insert that needs to be vented but there is no venting.this insert fits into a fake wood cabinet/ chimney. can i add charcoal filters and vent up in to chimney or do i need to add duct and direct the air back into kitchen?

I have one. Yes you duct back into the kitchen. They are worthless expense.

My vent in the face of the hood is hidden behind the woven rattan tray.