But with the circular blade of a track saw I wouldn't be able to get a straight vertical cut?
You finish the cuts with a fine pull cut saw, e.g. a Japanese saw. Any tidying up of edges is done with a sharp block plane and sharp chisels. Don't try using a jigsaw - unless you have a Mafell P1cc with the W1 blade the cut will be carp - jigsaws can't cut dense 40mm thick material plumb for toffee
A decent jigsaw with a decent blade will pretty much follow the groove cut by the circular saw.
Most people don't have a decent jigsaw - a sharp hand saw will cut better, a Hapanese saw bettercstill, but both will require some clean-up afterwards
I am not aware of a router bit that will leave a 2.2mm channel in a 40mm worktop.
There is no such beast. If there were it would probably break in the post, let alone if you tried cutting oak with it!
Even a belt sander (carefully used)
Then a trim router to finish the edges?
you might get away by missing out the “trimming” by going to the router after the cut?
A belt sander risks dubbing over the edges as well as taking too much material off
A laminate to trim the edge? It's 40mm thick oak that's a bit risky as you'd need to use a long, thin bit in a 1/4in collet. And why a router to trim the edges? A plunge saw with a 40 tooth blade would actually leave a relatively clean face on the oak which would only require light sanding with a sanding board to be good (basically a piece of 18mm plywood or MDF with some sandpaper glued to the face)
also are you happy with the hinge knuckle above the surface as sinking it below the surface will stop it opening fully as the wood above the pivot point closes up on opening
you can off course make a "V" groove to stop it binding
You are right about hinges, B-A. Strikes me that this requirement needs either bar flap hinges (done my fair share of those butterfly hinges - and the cut outs for the same) or something like Soss concealed hinges, or the circular versions of them (not sure what they are called - edit: "solid brass cylinder hinges") in which case accurate drilling to install them is essential
TBH this job doesn't need a lot of technology; it could be done with very simple tools by a - a cordless 165mm circular saw with a new 24 tooth blade (kerf c.1.5mm), a couple of straight plywood strips, some veneer pins to hold them in place, a Japanese ryoba saw, a sharp 38mm chisel, a sharp block plane and a home made sanding board...