Front brake causes violent shudder

Insufficiently rigid disc forks can always shudder. In this case it looks like there's a ton of bushing play. If someone wants to post an accurate treatise on how exactly the feedback loop is working, they would certainly deserve the win more, but: there is some kind of feedback loop from things flexing when you actuate the brake that causes the catch and release to happen at high frequency, which is another wording for fork shutter. You need a fork that sufficently transcends fork-shaped-object status to eliminate the problem. It would seem to not be the caliper's fault because all it's doing wrong is squeezing the rotor good enough to cause the problem. If there was something going on with the brake causing it to grab unevenly in different parts of the rotors rotation, maybe you could get it to do something like this, but that would be unusual and extreme.

Check for something loose. Something is moving or flexing.

Is the brake mounted tightly? Are all the brake components properly torqued?

Is the headset loose? Hold the bike upright on its wheels, squeeze the front brake to hold the front wheel in place. Push and pull forward and backward on the saddle. Does the frame move back and forth more than a very small number of millimeters because of slop in the headset? It shouldn't.

Are there any loose spokes? Check the tension on all the spokes on your wheels.

There's a good chance the shuddering is caused by something loose.

You have "play" in the system somewhere.

The brake is working fine in stopping the wheel when the bike is upside-down, but the wheel alone weighs far less than the rest of the bike plus you.

So, when the brake stops the wheel, it moves backward slightly, which flexes the area with play, changing the bike's geometry, which releases the play, and sets up an OSCILLATION that's what you're feeling instead of a VIBRATION

I suspect two main possibilities:

1. there is wear in the fork, so the lower stanchions move around inside the uppers. If the fork was not serviced for a long time (or ever), then this could be wear in bushings. I also observe rust on the chromed part - that's a sign of a long hard life and torn seals allowing water in. Your test of compressing the fork shows that it doesn't compress very much at all.
   Test for this by getting an assistant to squeeze the front brake lever, and rock the bike forward and backward. Meanwhile hold your hand around the chromed parts and the lowers. If you feel play, thats a cause or a contributing factor.
   A fork service might fix this, but could cost as much as a similar low-end fork new.
2. Headset bearings loose. The top and bottom of your heatset tube have a ring of bearings, the bottom holds your weight and the top one provides compression/preload. Again, this could set up an oscillation when under front braking.
   The test is the same as previous, but hold your hand around the stem, where it enters your bikeframe's headtube. There should be no/minimal play. If you feel it rocking like its two parts, then its too loose.
   Fix here is to tighten the preload.

Relatively unlikely causes of the oscillation:

1. I've seen this exact behaviour on an old rim brake bike, when the brake pads were super old and hardened. The disk brakes have a higher pressure and probably won't suffer the same way, but do take the pads out and visually check them for metal particles or strange wear.
2. Check for a loose front bearing by wiggling the tyre sideways. It shouldn't move at all.
3. Examine the frame for damage, like cracks. Lacking a structural member or brace could certainly give rise to an oscillation, but you'd definitely notice this.
4. Just confirm your front light mount and fender/mudguard is tight. They look fine, but they're also right where the problem is and any looseness can contribute to oscillations.

I agree with other suggestions that this bike needs to be fixed before you ride it again. As-is, a hard brake could act unexpectedly, and depending on the cause, it could get worse real quick.