Pros and cons of accessing sql server using ssms locally over on the actual server
As a general practice we never even install SSMS on the server hosting SQL Server.

You got the first two already 

1.Risk to accidentally rebooting the SQL Server.

2.Data security on the server.

However there are more:

First of all each session that goes to the server will take up resources (Memory/Cpu/Diskspace,DiskIO) needed by the SQL Server engine, this means a server might get under pressure by allowing people to directly connect on the Server. Which will slow down performance and might even cause timeouts to your applications.

When accessing the SQL Server Server, you can copy and duplicate data on the disks if there is not a good security structure, meaning you can easily fill up the diskspace where your sql server database files are located. Meaning your files won't be able to grow any more

If you have not set up the security rules well, you are able to install other software on the servers also taking away precious memory from the SQL Server server.