I finally fixed it
I took me several attempts, but here is a brief summary (as I noticed there are several threads on the subject, but only few of them present any solutions) that may help someone in the future.
* I tested some more for continuity and noticed both offending rows are shorted to ground
* this led me to thinking that maybe continuity between the rows is not the root of the problem, but maybe they just have the same issue
* I found some information about using multimeter set to resistance with range as low as possible to measure different continuity points and - by looking at the slight differences of resistance - figuring out possible short circuited area
* I only have a simple multimeter with lowest resistance range of 200Ohm, but it turned out to be enough. On row 2 there were 3 switch pads with 0.1Ohm resistance to the ground, on row 3 there were 4 such pads. The other pads had higher resistances (up to 0.7 Ohm, generally increasing the futher they were from the lowest-resistance pads)
* I examined the offending pads carefully and found out the cause of the problem. You see, I have a self-heating desoldering tool, which is very handy, because you just press it against the pad and press the "suck" button after the solder is melted (much faster than trying to do this with soldering iron and separate desoldering tool, the latter never having room to suck the whole solder at once). However, this self-heating desoldering tool has the "sucking hole diameter" slightly larger than the pad diameter, which means I melt the soldermask around the pad a little when I use it. Now, my PCB (not sure about others) has a copper mesh everywhere under the soldermask, probably serving as shielding. So when I melted the soldermask, it was enough to get very little solder between the pad and the shielding to short the row. To make things worse, in some places I damaged the soldermask so close to the pad it looked as if the pad was touching the shielding directly
* I took a small sharp knife and started creating "round" (not really) "borders" around the pads (see the photo at the end of the post). Once I did some of it and measured the continuity again, the short circuit was still there, but the lowest resistance value raised to 0.2-0.3 Ohm. I knew I was on the right path
* some time passed and I scratched around most of the pads with lowest initial resistance and the continuity disappeared
* almost there, but when I soldered just a few switches back, the keyboard become starting in bootloader mode and didn't go out of it. To make things worse, it didn't fix once I desoldered all the keys again
* after some debugging I figured out it must be Escape key, because it puts the keyboard into bootloader mode if it's pressed when attaching the keyboard
* I made several continuity tests and only after 5th or 6th attempt I found continuity between Escape column pad and the ground
* even though I couldn't see significant damage of the soldermask in that area, I did my usual scratching routine around that pad (somewhat more difficult, as column pads had paths attached on soldering side, unlike rows pads, so I had to be careful to not damage those)
* now the keyboard finally booted properly, I soldered everything carefully making sure the solder did not touch my scratches and now I can write this post on my fixed CA66
Thank you everyone who shared helpful info, indeed it turned into it was just damage, solder and shorts. I don't know how much time I lost trying to find the problem with the controller (I was assuming row 2 and 3 must have continuity via the uC).