Ok. The vent tube does exist (at least what I think is the vent tube which I'm assuming is a black rubbery tube that should run along the inboard side of the fuse box and goes down very close to a wire bundle). Nope, it wasn't connected to the Duralast battery that the truck came with. Also, the whole cover for the battery, fuse box and whole section in there is not is the best condition, seems warped and has a clip broken. So, it is clear that in addition I would need to work on the water ingress.
Thanks a lot for the tips and help! I will trace the corrosion and see if it survives and works after all the cleaning. If not, well, I think maybe will time for a new unit.

 

So, update! After cleaning the TCCM (I needed to re-solder a pin of one of the voltage regulators because it was in a very bad shape), cleaning the connectors, properly installing the plastic cover for the enclosure, cleaning the connectors in the actual box .... the fault seems gone (but I have not driven a lot, so fingers crossed!)