I was hoping that many Tesla enthusiasts would jump in on this one.  I haven't played with a Tesla coil since 1956, so I know  nothing.
But----  Isn't that little toriod  that is hanging off the side of the board supposed to be the "feedback" element?  Shouldn't the power wires or something make one turn through it, as in one of the pictures show in the "details" link in the aliexpress ad?
  I attempted to add a picture of the feedback setup on their earlier model, but I don't see it here.

Sounds like a reasonable secondary for a (DR)SSTC - you'll want to use reasonable fast IGBTs as it I estimate will resonate around 200-250kHz with a sensible size topload, but good TO-247 IGBTs should handle it OK (try FGY75N60SMD/FGA60N65SMD/FGH60N60SMD). There are some designs in this forum which could get you started with a full bridge PCB (order from a cheap chinese PCB house), and for a driver you could again use a design found here or get something off ebay/aliexpress (again, check the forum for reviews and options).

As you didn't mention having a toroid already, as an option I am happily using one of these with a secondary of the same 110mm diameter: https://www.ebay.co.uk/itm/Spun-Aluminium-Toroid-for-Tesla-Coil-HV-High-Voltage-300mm-by-75mm/113538948995
For that price it's not worth trying to DIY one! (Note comes in 2 halves, to hold it together either use the mounting bolt or roughen up the flat area on the inside and epoxy together).