2 major points from a purely technical perspective:

1.) Recirculating hoods are always worse than ducted ones - you have an extra charcoal filter that kills a major part of the airflow + makes it noisier.

Plus, people forget to change it and in a year or so it gets totally clogged, which makes things even worse.

Besides which, even if your homeowner is religious about changing the filter - the fact that it's recirculating, means all the heat & moisture is going right back into the kitchen. Ew.

You already have the duct installed - USE IT.

2.) Generally, the hood should be wider than the cooktop. Think about it - smoke spreads out, you want to grab as much of it as possible.

I do not even like wood hoods they get greaey even with good venting and they discolor over time from steam and grease . get a nice stainless hood you the ducting that is there .

Not venting is just gross grease and grime all over those new cabinets. IT's extra filth and smells for the house everywhere. Not venting is NOT an option when the vent is literally right there. So forget that.

You need a hood insert, and your cabinet maker needs to build a custom surround around the insert. Think 4-10K, depending on how fancy you want to get with it all. Or, just buy a nice metal hood already done from Rangecraft.

Generally it is recommended for hood to be a little larger than stove, ie 42” for 36” range for better capture. that being said i think this only really matters if you are using that duct - since you have one why on earth get rid of it?? one of the best things about my new kitchen was having an actual vent that works.

"I am going to do a recirculating vent and have the contractor remove the pipe at the top . This means the chimney will only go to the top of the cabinets ."

Who is advising you to come to such decisions?

Going with a recirculating vent is a really bad decision.

You definitely want to go with a 42" hood as the minimum width.

" Who is advising you to come to such decisions? "

Likely the same "designer" who designed those uppers to be so off in size.

Even though the vent isn't centered, you have the ability to do a nice hood surround that goes to the ceiling and covers the off center vent.

It's there, use it. And go at least 42".

Your kitchen looks lovely and high end - that really requires a functional hood. And, it will help keep your kitchen lovely and high end.

The hood should be wider than the cooking surface, I don't see how that is a problem at all. Having the upper cabinets the exact width of the lowers is the minimum. And a thing a lot of designers and builders do because it's convenient, not because it's the most correct way to do it.

There are plenty of rigid offsets you can get for your duct to compensate for it being off center and you have plenty of ceiling height for this to happen. The hood enclosure has to go to the ceiling anyway. The offset will be covered.