Treat the current hood like a decorative covering to all new everything internal. Like a standard wood hood. Or custom metal hood. That means buying an insert suitable to what it covers, and figuring out the vent size and fan location based on that.

Problem is the hood has visible buttons that I don’t know how to interface into a motor as they all have variable speed controlled by a potentiometer while these buttons interface with a board that then connects to motor.

Ignore the buttons and keep them as a decorative detail only. Inserts will have the buttons on the bottom.

Or get an all new hood. Because trying to reuse this is more expensive to customize to work than buying all new.

What you want to do is doable, but requires at a minimum an electronics tech, or maybe a precocious electrician to do it with minimum research and fuss. The potentiometer control establishes a voltage on its output as a function of (proportional to) the amount it is rotated. Out of the circuit, at any position there is a pair of resistances between the pot's output terminal and each of its other two terminals. One only need to duplicate these two resistances for each three positions and then use the H, M, L switches to select between the established ratios in place of the potentiometer.

Or, use a Fantech in-line blower of the type controlled by a (working from memory here) 10V dc signal. Then just wire the switches using appropriate resistors to supply one of three voltages for the blower levels desired. You will need a minimal 10V dc power supply to feed the switches. Fantech can probably supply a recommended circuit.

Moving the existing fan into the duct path cannot be evaluated from here, but I suspect it is too small if it was designed for a 6-inch duct.

Please, please, please, use duct sizes appropriate for the adopted blower's flow rate such that at full speed the flow rate in the duct is between 1000 and 2000 ft/min. Assume actual flow is 2/3 blower rated flow rate.

Last, I'm assuming that this hood has the necessary properties of adequate cooking zone overlap and adequate grease filtering (baffles). Blower size should provide 90 ft/min under the hood entry aperture, so actual flow rate should be 90 times the area in square feet. To provide this in the presence of pressure losses, a blower rating 1.5X this value is recommended.

And then there is make-up air to deal with.

Thank you for the detailed response. It’s a 600cm fan and the opening is around sq ft. So that seems ok (4x90x1.5=540). Doing some quick research the 6” will have a flow rate of ~4k but it drops below 1k when it enters larger duct which I prob can’t fix.

I did find this diagram for the wiring of my current hood and I don’t know if I can just wire a different motor

I would certainly want to increase the fan interface duct to get the velocity down to 2000 ft/min unless the length is very short. 4k is too restrictive -- pressure loss wise -- and ridiculously noisy.

Without the details of the PC board wiring, one has to guess. It is likely that even if you found a multi-wound external motor (some have been reported here; maybe Thermador) the board might not support any increase in power.

What is the blower configuration? By that I mean does it look like a squirrel cage, a propellor floor fan, an axial blower (jet engine look), or centrifugal blower (disk with blades like a carousel)?

In any case, Lomo and others above are providing the more practical advice -- go with a new hood assembly and external blower, preferably with an intermediate Fantech silencer. You can get ModernAire to build any shape you want, I think, and there are also suppliers of copper and other exotic combinations such as http://www.handcraftedmetal.com/. The only way you can save money with electrical hacking is if you can do the hacking. Alternatively, in some cases it may be practical to rehouse the blower further downstream if you can do sheet metal work.