What criteria were you using for determining decently matched pairs when testing with your 'old' tube tester ? Ways of determining if tubes are matched seem to vary across the internet - for me I would be looking at bias voltage, and anode current and then Gm.
Looking at test results in post #22 and ignoring the emission% column - which doesn't mean a great deal - I would think that tubes 2 & 3 and 1 & 4 would be a couple of reasonably close matched pairs. This is borne out when looking at the far right column which shows Gm in uMhos. Umohs was in common use on the more up-market US made testers of the 50s and 60s, 3800 uMhos is equivalent to a Gm of 3.8 mA/Volt.
As far as your old testers agreeing with the Amplitrex it's no surprise they don't, the Amplitrex uses Gm and the old testers use uMhos, in addition I'm guessing the old testers use lower than data book voltages during the testing process, which gives lower than data book results for Gm.
There is a direct correlation between the measured Gm (which is an indicator of the 'goodness' of a tube) and the applied voltages, tube tester makers could and did test tubes at lower than data book voltages - while it is a trade-off between performance and instrument cost, it is a perfectly valid way of testing tubes - the downside is that you get results that don't agree with data book values (or modern tube testers) but instead you need the tube tester manual/databook for the tube tester makers correctly calculated values to refer to.
Gm in uMhos is the only number I got from my old testers, so that is all that could be used for tube matching.
See Post #24, picture 1. It shows this new tester outputs Gm in mA/V and also calculates a Gm% based on the measured value compared to the specification for the tube.