Headlights don't work I pull the knob?

From the description it sounds like a bad headlight switch. The switch works by grounding the headlamps through a relay. As they work when you try the flash to pass then it isn't fuses or bulbs and cannot be the relay. It also cannot be the ground circuit as the lights all use the same one and you have working parking lamps. That leaves the section of the headlight switch or the connector on it as the failure point. A quick way to test it would be to insert a wire into the rear of the connector at pin D (Yellow wire) and turn on the headlights, no lights? Ground the jumper wire you added, lights come on? Headlight switch is bad.

That is the "flash to pass" commonly used in Europe, you pull the stalk back and it flashes the high beams to let the driver in another vehicle know you are passing them, not used a lot in the US, sometimes to warn of police or something in the road, but they include it anyway.
To get to the switch you need to remove the end panel from the dash, it just pries loose with a trim tool, then you will remove the screws that hold the switch in and remove the switch. The wiring harness will be right there for testing. The other item would be the headlamp relay but that should be OK as the lights do come on when you use the lever and the BCM controls the relay at those times.

Pulling the stalk that way is the flash to pass, it turns on the lights independent of the other switches. Both it and the main switch do that the same way, they ground the headlight relay and turn on the lights. The fuses all power the lights and the ones for the headlights are under the hood. One powers the marker lights as well as the relay. The other two power the headlights directly. The one above your finger is the fog lamp fuse. The one your finger is on is the fuel pump. The one to the left is the marker lamps, the two to the right are the headlamps. The headlight relay is behind the glove box in another relay box.
The vacuum display is a different circuit, do the other dash lights work and the parking lights?

From service info-
"Battery voltage is applied contentiously to the headlamps through two fuse. The HDLP L fuse for the left side headlamp and the HDLP R fuse for the right side headlamp. When the headlamp switch is turned to the HEAD position, ground is applied through the headlamp switch to the BCM using the headlamp switch output circuit. When the BCM receives the headlamp switch output signal, it applies a ground to the headlamp relay coil control circuit. When a ground is applied to the headlamp relay coil control circuit this engages the headlamp relay, closing the switch contacts and applies a ground through the headlamp relay to the headlamp relay contact circuit to the headlamp dimmer switch. Depending on the headlamp dimmer switch position the ground circuit would then be applied to either the high beam headlamps or the low beam headlamp turning them on".

I think the pull switch might be bad, but testing it by pulling in the switch itself out and testing the circuit at the back if it by jumping the yellow wire to ground would tell you for sure. If you pull the light switch out and the lights don't come on, then you just touch a wire between pin D and a ground and the lights come on, you just proved that the switch is bad and changing it would repair the problem. Attached is the schematic, that circuit for pin A to pin D is what the switch does to turn on the lights. If you pull the switch and it is bad, that circuit doesn't connect. The jumper wire then completes the circuit and if the lights work then, you install a new switch #10416123 If however the jumper doesn't turn on the lights then we need to test the next component which is the relay. However you say the flash to pass (pulling the dimmer lever) works. Which should mean the relay is working as that is how the lights get the power to work from.

Yes, there is a relay block behind the glove box that has the headlight, defroster, RAP and trunk lid release relays in it. When you pull out the switch the yellow wire gets grounded and that tells the BCM to activate the relay. The BCM can also activate the relay on it's own if you have the automatic headlight option as that just tells the BCM, hey it's dark, turn on the lights. Then you have the flash to pass which also sends a ground signal to the BCM to activate the headlights. The reason I don't think it's the relay though is because that last signal works. Basically all of the ways that turn on the lights use the same parts. The FTP grounds the yellow wire, that tells the BCM to turn on the lights, The headlight switch uses the same circuit to tell the BCM to turn on the lights. So in your case you can pull on the stalk and turn on the relay and the lights, but pulling the switch itself doesn't work. Think of it like a light in your house that uses 2 switches, either one can turn the lights on using the same wiring. Then one switch breaks, the other switch still works to turn on the lights though. I believe the pull switch is what has failed because both it and the FTP wires are physically joined at splice 221, however only one of them actually works. If the ground had failed none of them would work (including the marker lights that you said work) because they all use the same ground, same thing with the power as each side has it's own fuse for both high and low beam, as the FTP works the fuses must work. The wiring to the BCM and the relay works for the same reason. That only leaves the yellow wire on the headlight pull switch as a problem. So if you remove the switch and apply ground to the yellow wire and the lights come on, it's a bad switch.

Your just missing the 4th relay and someone has been there before because the tape isn't original. So you need to look at the relays and find the one with Yellow, Orange, White and Black wires although that likely isn't the issue. The part that's hanging down is the body control module. It is normally held in a clip between the steering column and power panel as in the third pic. You shouldn't have needed to take it that far apart though. The end caps come out of the ends of the dash and the switch bracket is right there with the cap out. The connector should have a retainer that you press then it pulls away from the switch. If it's that hard to pull it may be melted some and that could be why you have no lights. It would still need to come apart though if that is the case. Be VERY careful with yellow plugs most of the time those are airbag or safety connections and if they were unplugged it might be that the air bags have been deployed or bypassed. What color are the wires in the connector?

The white tabs get pried out a bit so they unlock from the connector, then it should pull out. Look at the wiring colors on the 3 relays you have, Yellow, Orange, White and Black wires on one of them would point to that one being the headlight relay. The missing one is probably for an option you don't have. The display is likely not related to the lack of headlights, it should come on regardless of the lights working. Most likely reason for that is a failed solder joint on the circuit board in the instrument cluster, or the voltage regulator on the board failed.

Normally the clips are pretty tight, What I do is use a thin tool to lift one side, then move the plug free of the clip, then go to the other side and repeat that so the plug pulls free. The clips to pull up are the yellow arrows, Just move them one at a time. Don't worry much if they break, if the switch is bad the new one has new tabs, if it's not the switch a zip tie around the switch through the plug would work. If you want to test it with it plugged in you can, you would take a sewing needle and slip it into the back of the connector so it makes contact with the terminal inside for the yellow wire, now pull out the switch, if the lights now work you found that the terminal in the connector is loose, they can be tightened if that is what you find, however if they don't come on, take a piece of wire connected to a good ground and touch the needle, do the lights come on now? If yes, you have just proven that the switch itself is bad. Replace it. Enjoy having lights again.
(called back probing a circuit https://www.youtube.com/watch?v=g2ST8w_0w-Q )

Yes you would ask about the cluster as a different question, that way if someone was looking for a bad display they won't miss it hidden in a post about a light switch.
However, being this is a GM you might want to go a look at this channel and search GM cluster to see what you can learn about the missing vacuum display, and maybe contact him about a repair if you like.
https://www.youtube.com/@jeffescortlx/videos