As sfdcfox mentioned in this answer that you can now Create, Read, Update, and Delete FieldPermissions via the REST API, you need to do a REST API call to insert the field permission for the newly created field via tooling API.
The FieldPermissions Object has mainly 5 fields ParentId, SobjectType, Field, PermissionsEdit and PermissionsRead. The ParentId field contains the Id of the PermissionSet. In Your case it is Id of the PermissionSet attached to the System Admin Profile.
You can do multiple REST API calls to get the PermissionSet Id from the Profile and the insert FieldPermissions or do a single composite request. The composite request Payload looks like this.
URL: /services/data/v54.0/composite/
Method: POST
Body:
{
"allOrNone": true,
"compositeRequest": [
{
"referenceId": "Profile",
"url": "/services/data/v54.0/query/?q=SELECT+Id+FROM+Profile+Where+Name='System Administrator'",
"method": "GET"
},
{
"referenceId": "PermissionSet",
"url": "/services/data/v54.0/query/?q=SELECT+Id+FROM+PermissionSet+WHERE+ProfileId='@{Profile.records[0].Id}'",
"method": "GET"
},
{
"referenceId": "NewFieldPermission",
"body": {
"ParentId": "@{PermissionSet.records[0].Id}",
"SobjectType": "Lead",
"Field": "Lead.My_NewField__c",
"PermissionsEdit": "true",
"PermissionsRead": "true"
},
"url": "/services/data/v54.0/sobjects/FieldPermissions/",
"method": "POST"
}
]
}