Since the ABS light does not come on does that mean that the connector is okay?
That's a good question. Usually, with an intermittent connection, the computer may think one of the wheels has locked up, so it displays the ABS light and you can hear it pulsing the brake on one corner - it sounds like a jack hammer noise.
If the car is equipped with traction control, the intermittent connection can also make the computer think a wheel is spinning - losing traction. (That's because an intermittent connection can open and close many times very quickly making the wheel appear to be spinning rather than locked.) So it will apply the brake, pulsing it rapidly. This sounds like a grinding noise, and is different than the ABS firing.
You may also feel this in the steering when it occurs.
I don't have experience with ESC - none of my cars has it. I think there are varying degrees of this which all fit under the umbrella of highly automated driving (HAD).
Anyway, long story short, if you have a bad signal from a wheel speed sensor, what I've seen is the car will disable the traction control (and presumably stability control). I'm wondering if that's what your lights are indicating, that it's disabling something rather than applying it?
The other thought is that a sensor other than wheel speed, which feeds the ESC, could be intermittent and causing your issue.
A scanner with ABS ability may help diagnose this. If it shows a wheel speed problem while the dash lights are on, that helps confirm wheel speed sensor.