Finally, I managed to repair my 2001, I think.
My meter had most of 200.x and 200.1 errors. Leaky old caps on the digital board (replaced) and somebody replaced the caps on the analog board around 2004-2007 (replaced those as well and the fan too). All the supplies seemed ok, but the errors were still there.
I found that resistors through which +-15V is supplied to the NE5534 on the ADC board are broken. One burned badly, leaving tar on the PCB, second one failed on high resistance/open. Replaced these resistors as well.
I've tried running comprehensive calibration, but it failed during open inputs step with error 354 = "200 mV offset out of spec". Then I've started chasing red herrings, replaced diodes in the power section, replaced the shift registers, some of the DG switches, NE5534 on the ADC board, desoldered the optoisolators and tested them on a protoboard - nothing helped. After all this messing around things got even worse and a dozens of new errors appeared.
But with TiNs help and some legal tender I got a new ADC board. With that I was able to run the calibration procedure with some poor artifacts (linear psu, 20k and 1M from a DIY decade). Things improved massively and I was only left with the errors that kada and SKE had: 200.1, 200.6, 200.7, 201.2 and turns out these can be fixed with upgrade to A08 (from A06). So once I got an UVC lamp (only had UVB) and a special PLCC44 adapter (the one I got with TL866 didn't work with 27C4096) I've upgraded the firmware and self tests passed.
I've started measuring some resistors that TiN kindly gave me, along with measurements from his 3458A, and I've noticed completely bonkers measurements on the megaohm ranges.
Now I wanted to improve things a bit, peroxide treatment (but the 25 year old paint on the display lens didn't survive kapton tape :/) worked nicely and while I was going to order DS1245 for MEM2 option I've ordered slightly better calibration artifacts, because I don't have anything top notch yet (multi-LM399 or LTZ1000) from Farnell: Y006220K0000T9L (20K 0.01 1 vishay-ppm/C), PTF651M0000BZEK (1M 5ppm/C), two ADR01BRZ to put in series and LT5400BCMS8E-8 9K/1K divider to divide ~20V to 2V.
And with these the calibration failed! Error 385, 20 MOhm 2W gain out of spec. High ohm resistors are a nuisance so I figured out that my soldering spree panic on the analog board left a mess affecting ohms current sources. After cleaning everything under a microscope, every guard trace, resistor cases, with dozens of cotton swabs and isopropanol the calibration worked!

I could even measure 1GOhm resistor.

Important lesson here is that self-test depends heavily on the calibration constants (which may be wrong or corrupted) and even these elaborate self-tests don't test everything that comprehensive calibration procedure tests. I still wonder what killed the resistors on the ADC board and worry that there might be something more wrong (a lot of the stuff is connected to +-15V rails).
I calibrated K2001 against my 3457A, as it seems most stable and extremely well aged (this one was closest to the resistance measurements done by TiN). Also, I really like this meter as it has most functions quickly available from the front panel, has offset compensation, 3 GOhm range, some autocalibration, no VFD but then I don't have to worry and it often runs 24/7 (also no fan) and does 7 digits in math mode - averaging. Unfortunately, it is not otherwise volt-nut friendly due to 3V front end arrangement. Transfer accuracy from a multimeter to a multimeter using some random voltage references wasn't that great, but none of my meters are calibrated anyway, it was more of a sanity check. Funny thing, when the GPIB cable is connected I get 104.1 error (GPIB handshake) but GPIB seems to work otherwise (well just *IDN? from ibtest, need to test more, luckily I have some NAT9914 chips at hand).
Now the meter occupied the last available space on my workbench and I want to build a more stable reference (multi-LM399 or maybe LTZ1000) and check how stable the meter is, drift, tempco maybe, etc. I also need to test some other functions (AC, current) more in depth to double check if there are no surprises that self-tests don't catch. Performance verification, even basic one, of such meter is definitely not an easy task.

Then if it turns out it is roughly OK I'll have to probably calibrate (low level calibration) this meter. I don't feel like spending more on the calibration than on the meter, also Tektronix has been annoying me for a long time, either once I wanted to buy 2000 benchkit parts and they gave me silly lead times, or other times I had to create an account there, as I was downloading TLA stuff or I wanted to download a datasheet for an SMD oscilloscope probe that I got off ebay. It was enough to trigger their marketing department and I started getting annoying phone calls. Now they stopped calling, but I still get emails telling me to buy TBS2000 or other crap - they seem really desperate. Anyway I couldn't even find UKAS Schedule of Accreditation for Tektronix (not that I need certificates, I was just curious on the DC 10V specs, Keysight UK has 0.8ppm). I'll ask RS Calibration for a quote. On their price list it says 62 GBP for a bench meter. They might not do 7.5 digit, but at least they could transfer whatever they have currently in their Fluke calibrator - better than nothing. I hope they have GPIB scripts for K2001.
Overall, I quite like this meter. Maybe not as much as 3457A, as I have to go through multiple menus and the average display from readings stored in a buffer are using engineering notation with only 4 digits after the decimal point, but on the positive side it has big VFD, sensible ranges with GOhm impedance up to 20V, thermocouples and similarly to 3457A offset compensation as well as scanner card option.
I would like to thank the forum members for helpful posts here and TiN in particular for providing me the ADC board, as well as putting a lot of effort on writing articles about his repair adventures - it probably takes as much time to write and take pictures as it takes to repair some things.