Well, I made an attempt today, and I can kinda confirm my secondary suspicion, that the primary cause of the issue is an incomplete reset circuit on the NetUSBee Ethernet part.
First I rigged up a 22K resistor (the specs say 27K, but I had no such resistor at hand) to pin , but that didn't solve the issue. Regardless, I found an Open Source ISA 8Bit 8019AS based card project that does put a 27K resistor on that pin
(see R6), so probably it is indeed best practice to do so.
But this sadly did not solve the issue, with the original STE PSU, uIPTool still didn't work, NETUS.TOS was still failing all Ethernet tests.
So what I did afterwards, I rigged up a 1K resistor to the RSTDRV pin of the RTL8019AS, and made it short to VCC during power up, essentially emulating the ISA RESET signal. As the machine was booting (Atari logo appeared, TOS 2.06), I disconnected the resistor from VCC, so the card's 10K pull-down could take over, and take the chip out of reset. Then after the machine finished booting, I started uIPTool, and it worked immediately, first try! I could browse the Atari's content from my Mac, so it was actually working, not just appearing so. Basically I was never able to get the Ethernet part running with the original PSU, until now, and here it is, fully up & running!
So this concludes I guess, to a large extent that at least some of the various "finicky-ness" of the NetUSBee in various Ataris and with power supplies is an incomplete reset circuit. Thanks to the RSTDRV line simply grounded, the chip starts to initiate a power-on, while the power isn't stable yet, and it will end up in a catatonic/half initialized state, where it just doesn't answer to the bus.
Now, I'm not a hardware engineer, so it's entirely possible that the issue could be worked around with changing various decoupling capacitor sizes, and whatnot, and not require a full-on reset circuit, but my hardware-foo is way too weak for that. Please advise.
What I would do if I had a chance of redesigning the card, to put a flip-flop on the ROM3/4 select lines, then connect that to RSTDRV, to keep the card in RESET, until those pins are first asserted as the OS during boot looks for the expansion ROM. This is late enough so the power can be 100% stable, but early enough so it shouldn't cause any issues with real drivers. But maybe someone has a better idea.