I had a nice talk to an engineer from Keithley Germany today after their distributor forwarded my questions to the Keithley tech support.
My interlocutor obviously had very deep knowledge of the circuit in (not only) the Keithley 2000 multimeter and took time with me to go through my questions and do some tech nerd chat - a quite pleasurable experience and a very positive surprise, given the negative comments here about Keithley support after they were taken over by Tek!
In short, as neither leakage current nor bias current of the Keithley 2000 are specified, the failure of the self-test steps 306.2 and 306.4 are not considered a defect. This is true especially if the meter was upgraded from an older firmware which did not contain these tests. That applies to many (if not all) meters which show the error at my employer.
If the "bias current" steps fail, this usually does not mean there a problem with real-world measurements - as long as the meter passes the calibration. The user should be aware that these universal DMMs are not electrometers, and low-voltage measurements at high-impedance sources can be influenced by these currents. 100 pA bias current should be considered a "typical" value, even though it may be higher at some older meters and much lower at some new units.
When we try to measure the bias current, we should be aware that the current depends on the voltage applied to the terminals. When we just connect a 10 MOhm resistor to the inputs, we get the pure bias current. When we add a voltage source in series to that resistor (like the self-test does), we get a leakage current in addition to the bias current. Both of these should not be confused with the input resistance, which is usually somewhere around 100 GOhm.
I was also told that newer revisions of the meters contain some optimizations which lower the leakage current, and the leakage current should be constant over time and should not change much when today's new meters get older. The correlation of multimeter age and bias / leakage current is very likely not only because of component ageing, but also improvements in parts (and maybe PCB layout, but he did not say that).
The internal self-test was rather intended for service at the factory than for the end user, and it sounded a bit like there are regrets that the test was made available on the front panel. As a consequence, that test was "hidden" when they later created the 2010, which is quite similar.
I hope this helps!
I might still do some experiments with the NXP JFETs instead of those from On someday, even though I am aware of the scattering in parameters. However, repair of my Adret 103A calibrator is given priority over that now, as the self-test errors on my Keithley are gone and the remaining leakage and bias current are not critical.