web search engine indexing

I’m trying to use this plugin for a personal blog, I wonder if the google-search-engine will still index my blog site, because behind this plugin all my posts are still public by default. It will be a surprise if I can google my content which is supposed to be private.

Additionally, is there a way to remove the W icon(links to wordpress.com) from the login page?

https://ww.wp.xz.cn/plugins/jonradio-private-site/