Thanks Gagan! This did the trick! Turns out I was being too clever with config and all.. After moving the wordpress install to blog and making just a few minor modifications, I was able to get the site working properly.
This could be a problem with your webserver configuration.
Does visiting https://example.com/blogs work for you? If it also redirects to https://blog.example.com/ then take a look at your webserver config. There is probably a config doing a rewrite / redirect.