Keyword not supported: 'statslog_on'
It looks like SSMS is using the .NET System.Data.SqlClient API.  As far as I can tell, the statslog_on connection property is not supported using this API.

I ran a simple program* that iterates over the key value pairs allowed by that API and came up with these keys that are permitted in connection strings for SqlClient connections:

Data Source
Failover Partner
AttachDbFilename
Initial Catalog
Integrated Security
Persist Security Info
User ID
Password
Enlist
Pooling
Min Pool Size
Max Pool Size
PoolBlockingPeriod
Asynchronous Processing
Connection Reset
MultipleActiveResultSets
Replication
Connect Timeout
Encrypt
TrustServerCertificate
Load Balance Timeout
Network Library
Packet Size
Type System Version
Authentication
Application Name
Current Language
Workstation ID
User Instance
Context Connection
Transaction Binding
ApplicationIntent
MultiSubnetFailover
TransparentNetworkIPResolution
ConnectRetryCount
ConnectRetryInterval
Column Encryption Setting
Enclave Attestation Url


This list was produced specifically using the .NET Framework 4.7.2.  The list could be different depending on the versions of things that are used by different versions of SSMS.  

For instance, I have SSMS 18.4 installed. Attaching a debugger tells me that it's using the .NET Framework 4.8 version of System.Data.dll:



Based on that, I would expect any of these to be allowed in an SSMS connection string (give or take what might have changed between version 4.7 and 4.8 of the framework).  Of course, some of them might fail at runtime, might not be compatible with one another, or might depend on the type of database you're connecting to.

Some allowed aliases for those specific values are allowed, and documented here: SqlConnection.ConnectionString Property - Remarks

*here is the C# program I used to produce those values:

var builder = new SqlConnectionStringBuilder();
foreach (var key in builder.Keys)
{
    Console.WriteLine(key);
}
Console.ReadLine();