So I've been trying this again, and I have a plausible explanation. Apparently the ESP01s had pull-up resistors on three pins, including the ch_pd. This is a 12k resistor, which means that when I connect to ch_pd to ground it draws 3.3v / 12kohm = 0.275mA! The ESP01 (without the s) does not have this pull-up resistor, it's only there is the newer s version. I'll try to de solder it and see how that affects it.
ouch, BTW it happened to me too, fortunately I found pack of 20 fuses for my multimeter (10 for 600mA and 10 for 10A range) on aliexpress for ~2USD so I am ready when it happens again :-)
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So I've been trying this again, and I have a plausible explanation. Apparently the ESP01s had pull-up resistors on three pins, including the ch_pd. This is a 12k resistor, which means that when I connect to ch_pd to ground it draws 3.3v / 12kohm = 0.275mA! The ESP01 (without the s) does not have this pull-up resistor, it's only there is the newer s version. I'll try to de solder it and see how that affects it.
(yes, I bought a new more advanced multimeter)