I'm happy to report I managed to fix the board. All the ICs (except U12 and U3) were replaced not only once but twice!
After ensuring the drivers don't feed high voltage back to the controller I replaced it along with the ESP. After I flashed them both I was getting excited seeing working web control. I tried the drivers but no response. So I assumed they are fried too. Replaced all of them (the board thermal characteristics is soooo freaking good - unfortunately in this case ). That didn't help, still no working motors. I was too lazy to turn the oscilloscope and check the signals. I assumed the only chip I didn't replace (U10) was faulty and ordered a new one. Today I replaced it and still no progress. Only then I checked and found SPI data wasn't getting due to poor soldering.
So I removed the controller and re-soldered it with more care. And voila! All good. So just to complete and to avoid any hidden soldering problems I replaced the drivers to the old ones, which as suspected work just as fine.
Here is the board working and looking like new again