PostgreSQL replication with periodic log shipping
This was most probably an issue on my part. I decided to do everything again, but this time making a full streaming replication following the excellent guide here: https://www.youtube.com/watch?v=Zq4zS0ZbXsc

Then, I stripped this solution from the streaming part (removed connection from hba file and a line from recovery.conf). Just in case someone would want to achieve a similar solution - do not compare WAL files by their 'Last Modified' date! You are interested in copying WALs newer by their 'Create' date. This is due to the fact that PostgreSQL creates some files and then for its own reason changes their 'Last Modified' dates to the ones in the past.