Yes, that looks like one of them, pretty sure there were 2, one more blocked than the other.
Other one may bee was a breather for the gearbox.

I would start by checking the battery terminals are clean & tight, a loose connection might induce a voltage spike from the alternator.

Do you have a Lithium battery?
They have their own battery management system ( BMS ) inside that tightly controls the battery cell voltages.
Lithium cells don't like being either over or under their regulated voltage ( will be permanently damaged ), which is why they have a BMS.
If the battery voltage rises too high the BMS can just switch off the battery to protect it from overcharge, this again could cause a momentary spike from the alternator.

If you have a spare battery, try that.

I was thinking on similar lines to Rich.
Perhaps one large dia shim for the OD & a smaller dia for the ID.
https://www.ebay.co.uk/itm/224326415431?chn=ps&var=523117972233&_trkparms=ispr%3D1&amdata=enc%3A1q_ZHns0AR8GigKvC1eQ7oA60&norover=1&mkevt=1&mkrid=710-134428-41853-0&mkcid=2&mkscid=101&itemid=523117972233_224326415431&targetid=1647205088320&device=c&mktype=pla&googleloc=1007000&poi=&campaignid=17206177401&mkgroupid=136851690655&rlsatarget=aud-1195034332244:pla-1647205088320&abcId=9300866&merchantid=6995734&gclid=Cj0KCQjwk96lBhDHARIsAEKO4xYpbxiYrrrWcp5JtMdB2VRupgAUOqgn49dQmBgVwFMGh0hxQGlbizUaAjs8EALw_wcB

The sump plug certainly doesn't look healthy but also depends how many miles since it was last cleaned.
The oil is shared with the gearbox so that can explain some of the debris.

The fault code printout show the model as being R1200C, so may not be giving the correct codes for a K1300.
Try to get it connected to a GS911 & see what codes that uncovers.

The knock sensor is just basically a microphone attached to the block to detect the onset of knocking in the engine as the ignition is advanced. Will probably just need a new sensor or the wiring/ connector has been damaged.

The GS911 might hopefully give more info on the gearbox sensor fault.

I assume the Exhaust Gas Throttle is the flap in the exhaust just before it connects to the silencer.
This has a habit of sticking closed & causes very poor running & loss of power.
When mine stuck closed the bike was gutless & engine would stall easily.

If you take off the cover at the back of the flap where the operating mechanism is, it can be wired permanently open. No more problems from it after that & no detriment to the engine running at all.

Let us know how you get on.

" I was wondering how the drive flange might have become distorted myself. "

Isn't this the flange that was part of a BMW recall some years ago.

I remember mine getting changed. It involved heating up the new one to get it on then making sure it was seated correctly.
I suspect it probably wasn't seated correctly when fitted, proving that would be impossible now.

Never the less the selling dealer is completely responsible for correcting the issue at no cost to you.

Your cheapest way forward is through the small claims court which I think costs about £50.

For the dealer to defend it in court would cost more than the repair.

That might just persuade him to do the decent thing & repair it.

And hideously expensive to repair when it all went base over apex, which most of them used to do fairly regularly, my dad had a couple of DSs and eventually gave up on them as it needed specialised surgery to fix as opposed to just popping on a new 'shock absorber'...
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I never owned a DS unfortunately, still think it is a great looking car.

Had umpteen CX's. The only thing that used to go wrong with the suspension was the height corrector linkages located under the car used to seize up occasionally due to exposure to all the spray & crud thrown up underneath, just needed a spray of grease & they were fine again for another couple of years.
The suspension spheres that provide the damping on each strut needed re-pressurizing with nitrogen every 3/4 years, that was a 15 minute job by a specialist @ £15 a strut back in the 80's.

Have to say I don't think they really cost me any more than a conventionally suspended car.

Rolls Royce used to use the Citroen system in their cars, need to say no more.

" If you could take the spring off and just have the damper working in its normal range (which you can't because the whole thing will just collapse till the damper bottoms out) you would get a very jittery and jarring ride because the damper can't absorb the shocks. "

Citroen have been using this system for years on some of their better models.

Their hydro-pneumatic suspension has been in use since their DS model in the 50's.
It continued with their GS, CX, C5 & C6, not sure which model has it now.

I had 3 of these models & have to say they were the most comfortably suspended cars I have owned.

It's also a self-levelling system & can be used to raise the car to change wheels etc.

Probably a bit more expensive to manufacture.