In our schools database page, we have filters defined to search them.

There is default filter feature at the bottom for "Show my schools only" which I think is for displaying only the Author's records.
Since we are not allowing any non-admin users to add records, there is no need for this button.

Is there a way to toggle this checkbox off?
Or where may I find it in templates that I can remove it?

I know this is not the "right" way to do this, but I can hide it using js:

Theme > cms > databaseFilters

Add this at the bottom:

<script>
  var el = document.getElementsByName('cms_record_i_started_checkbox');
  if(el[0]) {
    el[0].parentNode.parentNode.style.cssText = 'display:none';
  }
</script>