I see you have gone for the scatter gun approach and spread your issue under different headings just to confuse the likes of me.. (-:
Being an old geezer I kinda gave up on messing around cars in frustration when the digital era arrived and black boxes took control along with the need to swap out expensive parts to fault find..
Assuming anything in modern circuitry seems to risk a degree of confusion, so perhaps best not to..?
i.e Lighting circuits may not be fed via the ign switch..?
Back to basics thinking....? You have checked the lamp filaments of course, assuming they are not LEDs, and if LED there seem likely to be driver modules involved...?
In times past the main, dip and flash stalk is generally designed as an either/or switch with a further function to flash the main beam... also the horn would be activated by pressing the end of the stalk towards it`s base. EITHER dip OR main beam seems kinda obvious electromechanical action as does the action of the switch to flash the headlamps..?
I suspect you have proved that the switch moves freely to all those positions and holds to the either and or positions to enable selection of main and dip..
It seems probable that there may be a local plug to enable simple replacement of a faulty main/dip switch...?
If there is such a plug, it seems that the feed/live/hot, should be supplying as it should if you have any headlight circuits operating... But check it is there to be sure.
The fault could be in the plug it`s self check for signs of heat in the wires and the plug (if there is one such)
Having confirmed there is no signs of melting and or burning at the plug and that the there is a voltage available at it....
That being the case with the plug disconnected then you can hopefully use your multimeter on ohms scale or buzzer to determine that all the above switching actions are taking place within the switch, making use of the wire colours to identify and determine primarily the feed/live/hot into the switch, then operating the switch to determine that the mechanical action is creating the appropriate electrical connection, again identified by using the wiring colours to confirm all is well for each and every individual action...?
OK so with luck you have proved there is electricity available to the switch and that the appropriate switching action is taking place within the switch..?
That being the case then it is possible that when plugged up and passing POWER through the switch which exceeds that of the multimeter buzzer etc... The switch can fail under LOAD due to burned or damaged contacts.
The above scenario was oft times created when folk fitted higher powered headlight bulbs drawing more power than the switch was designed to handle and thus in time the switch contacts would slowly overheat and burn creating electrical resistance across the internal contacts where carbon would build up reducing the contact area..
I suspect if you have got this far then your pin method used through the insulation may prove that the switching action it indeed functioning correctly by switching the power through the plug and on into the rest of the wiring loom..?
With that proven where to now..? There may indeed be relays involved or not..? OK so you have a jumble of wires and a multi-plug base to confuse matters, however with luck you should be able to pick up whether or not the OUTPUT wires that you have proven should be switching as required, are to be found entering the multi-plug base...?
If not then chances are as in times past there is no relay in the circuit.
I guess another possibility may be... Now that you have hopefully proved the switching action, with the headlight switch selected on and switching between main and dip, you might hear a relay clicking to assist in proving if one such is in circuit..? Or id your hearing may not be up to the task, then if you place your finger on each of the relays in turn as you switch from dip to main or vice versa..?
Of course if no relay action heard or felt, the connections to/from the coil of the relay, or the coil it`s self could be open circuit.... Thus proving just how important it can be to make good use of the wiring colours to identify circuitry at the base of the relay and elsewhere..?
Yeah all a tad basic I suppose but then hopefully my old way of thinking might help and not just add confusion.... I`m oft times confused myself thee days..

BEWARE...I have not read over this so there may be flaws to be found..
