@filipm the EARs transformer is pretty noisy mechanically at your place; suggesting a power issue. The battery test was interesting and you would think would rule out dirty power, but that may have also just demonstrated the same power issue - have you tried the battery test elsewhere (like at the friends place where the EAR sounds fine)? I suspect the same results.
I listened to the noise again - the noise is a pulsing sound over what looks and sounds like a ground buzz. This may be a stretch, but perhaps your EAR's power supply is near end of life and when combined with the transformer issues at your place, is sagging and recovering rapidly, like a motorboat noise. It may not occur elsewhere because the wall power is just better enough there to allow the EAR's power supply to keep up. This would cause the tube circuit to generate the noise in the output regardless of input (search tube motorboating on youtube).
Are the filter capacitors in the EAR's power supply original? They are probably over 20 years old at this point if so. I'd take those out and test them or just replace 'em.