Fix to Grayed-out Safari Homepage Setting
Become able to change your Safari homepage:
(Un-gray the setting in Safari Preferences)
1. Create a new Administrator user account.
2. Log on to that new account.
3. In System Preferences, for your main (problematic) user account, uncheck Enable parental controls.
When you log back on to your main account, you'll find that you're now able to change your Safari homepage. And then to keep things neat, you can delete the new user account that you had created to solve the problem.
Background
This issue happened to me after I downloaded SafeFinder for mac in what was disguised as an Adobe product. I went through all the recommended fixes like downloading (and running) Malware Bytes; deleting a Profile in System Preferences; deleting Safari's extensions, plugins and browsing history; and deleting select folders in Library. Nothing worked.
It was the final comment here that pointed me to a solution, so thank you to user PureShores!
It turns out that the problem is caused by clever trickery via the parental controls.
System
Mojave 10.14.4
Safari 12.1
Hope this helps!
MacBook Pro 15”, macOS 10.14