When braking between 80-60 kph the steer is

great danger there. dont go fast.
can be tires out of balance (tire weighs feel off)
or wheel falling off (ever check the lugs)
tire bad. (bulged, delaminations , man killer tires, like Firestones)
tire rods balls, ready to fall off (more man killers)
ball joint ,ready to fail. and far far worse...
why not get the car serviced, have them check it out before.... bad happens.
noise and vibration or shaking on any machine means , doom is coming. in most cases,.

warped brake rotors.

Sounds like a warped rotor or uneven rotor surface,could also be a worn or damaged wheel bearing.

Check tyre balance (see a tyre dealer)
also get check wheel alignment, front wheel bearing adjustment and that the tyres haven't got tread separation.
Then if not any of those show a problem, check tie rods and steering rack ends.

You need to have your front rotors machined or replaced with new brakes.

OK, when the ABS module is unplugged, the same results. That tells me that the ABS system is not working properly.
Did they look into the master cylinder and other abs related components?
Any ABS codes on it?

If steering wheel shakes above 50 mph that's likely to be the tires on it, rather than a brake problem.

Have tires balanced and rotated and inspected for out of roundness that might cause vibration above 50 mph.

a steering wheel shake with the wheel fully turned usually indicates a seizing constant velocity joint (axle). it can also be accompanied by a clicking noise while turning at higher speed. this is easy to confirm. raise and support the front of the vehicle and put in neutral. turn the wheels by hand with steering wheel pointed straight ahead. there should be minimal resistance. then try rotating the wheels with the steering wheel turned all the way to one side then the other. if you feel any extra tightness or binding, you indeed have a seizing joint. if the axles look ok visibly, you will need to remove the axles to confirm which side is the problem. good luck!

check the resevoir for fluid level and check the tension on th p/s belt.

My immediate quick test would be to move the front tires to the back (and backs to front, of course) then try it out. If the vibration is still there but feels like it's coming from somewhere else (vibration could feel like it's in in steering wheel or seat), you've got a broken belt or bent rim (wheel). Don't want to get your hands dirty? Get all 4 wheels balanced -5 to 10 bucks per wheel.

The problem lies in the front rotors. If the steering wheel is shaking without braking, then we need to break this up. If it's low speed(1-15 mph) then you have a tire that is coming apart. Highway speeds, you need the wheels balanced. If it's only while braking then you need to have the brakes inspected. If pad replacemnet is required, go for it. When they address the rotors, DO NOT TURN THEM ON A CONVENTIONAL LATHE. Honda REQUIRES them to be either turned on the vehicle(on the car lathe) or replaced. Then it comes down to quality. Always replace with the higher quality parts on these. If you do not, the problem will occur again, and the sme is true if you turn the rotors off of the car.