I was having the exact same symptoms. After futzing around a while, the culprit turned out to be the fact that my blog address was located in a subdomain while wordpress install dir was not:
wordpress install dir:
<https://www.mydomain.com/wordpress/>
blog address:
<https://blog.mydomain.com/>
The problems finally went away after I moved my wordpress install directory under the subdomain:
<https://blog.mydomain.com/wordpress/>
Now “Site Admin” link appears instead of “Login”, and “Edit this entry” appears on each post. Heaven! Hope this helps.