That thing is a useless joke and should stop ever being sold. Seen it dozens of times on here, and it is just an irresponsible product to ever sell someone. It will not work for proper venting without butchering it so that a real full depth insert will fit. That is just a silly fire hazard as it exists.

Definitely not music to my ears, but thank you for your response. Can you help me any further in this design and how to achieve some thing that well be functional, attractive and safe?

You need to have a rigid duct, and the diameter will need to be much larger than that flex duct.

In a custom hood you will need a metal liner, which is sometimes custom. It looks like a shallow box, usually, and then the range hood insert fits inside the custom metal liner.

Any vigorous cooking will require a baffle type, not mesh type filter.

The hood entrance aperture, see obvious lighter edge area in image above, needs to overlap the cooking pots and pans by at least 3 inches all around.

Achieved full power flow rate needs to be about 90 CFM per square foot of entry aperture.

Rated blower flow rate needs to be roughly 1.5X the value calculated from the requirement above.

Duct diameter need to be such that the velocity in the duct is in the range of 1000 ft/min to 2000 ft/min.

For less noise a silencer can be used between hood and blower (if remote).

Make-up air (MUA) is needed for the hood to work. It is normally supplied by an MUA system, the scope of which depends on whether the residence has combustion appliances without their own MUA, but in most municipalities, at a minimum provides at least as much air as the hood is expected to remove. (Hood air and MUA are always equal, but this may mean a gagged hood.) Depending on open windows or cracks in the walls generally won't do.

Thank you for this feedback and your honest comments. You are helping me to work through next steps.

Go to a good appliance store that installs what they sell. Not a big box store. Take your pictures with you. They will sell ventilation products and know how to install them properly.

Get rid of the laundry room flex ducting. Rigid metal duct needs to be used. This stuff is a fire hazard.

The fact that a builder would use that type of ducting product says that he's cut corners in all sorts of places or he has no eye for detail and supervision.

“Get rid of the laundry room flex ducting.”

Actually, the flex duct is against Building Codes for dryer ducting. That is because the interior multitude folds of the flex duct collect lint creating a very serious fire hazard. Lint buildup, plus the heating system inside a dryer, is a recipe for fire. Similarly, a flex duct for a cooktop exhaust allows grease to collect in its folds and creates a strong fire risk. Both a dryer exhaust and a cooktop exhaust require rigid metal ducting.

I agree that whoever installed the flex duct in the OP’s kitchen is ignorant and incompetent. Do not continue to use that person.