How to resolve ORA-01031: insufficient privileges error on Windows installation?
The problem is that your Oracle Client installation is in the system PATH before your Oracle Server installation (in fact, you even mention that it worked before you installed the client).
Proceed as follows:

Remove the Oracle Client as it's only confusing things. Use the uninstall option but afterwards check that the directory and the Oracle Client registry key is gone and remove by hand if necessary.
(It's possible to get this working by having the Oracle client home in the path after the server home, but it's much simpler to just have the single install. An Oracle server installation already contains a complete Oracle client anyway).

Make sure that the path to the server's home is in your PATH. And no other Oracle home is in the path.

Make sure that your account is in the ORA_DBA group. Double check this, especially if you are using domains.

If all else fails, you could remove SQLNET.AUTHENTICATION_SERVICES=(NTS) from your sqlnet.ora as this will force you to use password authentication.