Accuracy aside, problem is all your test gear are built for other uses are not meant for verifying a 6.5 digit DMM. The Fluke 1586A has too narrow a range to be used as a full range transfer standard for the 34401A as it can only go up to 50V and 100mA. Your Fluke 707 can only source up to 20mA with 0.015%, and the 724 is only good for 10V at 0.02%. They are all below the accuracy level of the 34401A by a large margin. The gain verification on the 34401A has verification points at 100V and 1000V, as well as 100mA, 1A and 2A, which you don't have a way to test. So any adjustment is definitely out-of-the question.
For the ranges you had checked, they are out of spec but not terribly off. So it all depends on your use case on whether it should be adjusted.
I own two 34401As, but they are both late Agilent builds before they changed to Keysight, so I cannot comment on the drift characteristics of the older HP meters. For my meters, they do drift a little early on, but then they become very stable. In your case, an older meter without any cal history (or who cal it), my guess is it would drift very little now, but could be out-of-spec as your test shows. If you use case dictate accurate 6.5 digit reading, it is a meter that is well worth a professional calibration as it could give you many more years of reliable service (assuming it is in good condition and properly maintained), and delivering an accuracy level none of your current equipment provides when they are used as a DMM.
The Cal Count in the meter should also give you an idea of how many time it has been calibrated in the past (thought not when unless they set a message).
For the early HP meters, if it has not been modified, all the inputs are red, the later ones are red and black before they move to Agilent. Also the early units were built in the US before they moved production to Malaysia (and even then, the components were made in the US but assembled in Malaysia early on, before they moved component production over there). If it is a valid serial number, my guess is the 31 prefix would make it one of the early production with '91 design (the meter was announced in '92), there is also a possibility it was built the 46 week of '91 in the US, depending on how serial number is applied to this unit. If it is the case, the inputs would be all red. If you open the meter, you may also find more date information on the PCB or chip as well. In any event, it is very likely it is 27+ years old.
If you look at the service manual, you will realize that it is very difficult to do any cal on the 34401A unless you have the requisite calibrator. There are just too many verification points that requires high level of accuracy. It is not something that you can do reliably without the proper equipment. For the data you provided, I would either leave it alone, or have it done professionally, but not to try any adjustment with the equipment you have.
I often take the abundance and the low calibration cost in the US for granted, as there are shops here that would do a NIST traceable cal for less than $100. I send my meters back to Keysight for cal yearly, it is $240. You may want to check with your local Keysight office to see what they quote you, but I doubt they will charge you 400-700 euro.
I still use my 34401As regularly, they are accurate and reliable, and great 6.5 digit meters.
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