I'd have to see the schematic of the cooktop to know for certain but offhand I suspect that it is designed with very many coils over a broad area to create even cooktop heat, but this means a higher working voltage and more coil turns, so if you just use a few turns you are going to put more current through them than the controller can handle.
Frankly an induction stovetop is not a good way to start the project. Too complex and set up for a high voltage that should stay enclosed. My vision for this would be something like the following:
Pick an 18V cordless donor tool, compatible with a brand /batteries I already have. Gut it except keep the trigger as an on/off switch. It needs to be a tool with sufficient space inside to house the circuit board, and might be nice if it had room to stuff an LED driver board in it and mount an LED pointing at the work.
Pick a far more basic inverter board with matching coil included. For example the link below is one or search through many on ebay as "induction heating module coil", seeking one that at 18V or more (12V cordless probably can't provide a satisfactory wattage output), has a current spec within what typical cordless tool batteries can do for short duration, say 20 amps. You might easily find this example (or one even better suited) cheaper on another ebay listing or elsewhere:
https://www.ebay.com/itm/New-DC-12-...ard-Module-Coil-Fan-DIY-Kit-S0M3/303156284610
All these are set up to mount the induction coil to the PCB but you don't have to do that. You could mount the coil on an extension of
flexible metal pipe (or tubing) and run a couple of low(ish) gauge, high strand count, silicone insulated (highly flexible and heat resistant) copper wire to the induction coil, allowing it to get into tight spaces. Run a couple more wires if you want the LED remote mounted near the coil.
If the cordless tool trigger switch is low current rated and just biases a transistor for motor power, leave the transistor/circuit in and take the output from that to the induction board. This will only necessarily work with brushed tools, brushless would depend on the design and reverse engineering it, and you don't really *need* variable trigger control, just an on off switch to run at full power and then stop when it's hot enough.
If the budget stretches higher than the above example and you're willing to have a mains AC power cord to it, there are other examples on ebay of those, but do take care to properly insulate it if dealing with lethal (above say 36V would be a conservative limit) voltages. You could keep that safer as a homebrew project by having an external brick AC/DC PSU tethered to the tool, something like a 24VDC, 20A PSU can be had for $18 delivered on ebay.
https://www.ebay.com/itm/AC-110V-22...h-Power-Supply-Adapter-LED-Strip/352657328504