Just want to write how I solved it and the outcomes.
Since verification process only checks a local file, of course it didn’t pick up DNS records.
In the end, I solved the problem by adding the requested file manually to the location where verification process was checking.
All you need to do is to create said folder and file. Then add the code you get inside that txt file. After that run the verification and voila.
I really don’t know why they force the depreciated plugin. I also don’t know why there is no simple guide explaining these processes for dummies.