It sounds like you've found the setting to enable Institutional scoping, which is a good way to implement what you're looking for:
One note about this: since the setting was added, we've also added the ability to link an authority record to a repository. If you have associated your authority records with maintaining institutions, then it's also possible to use the Menu module to add a link to the institution browse drop-downs that will show authority records related to the repository. Here is how I added a new node to the Institutional browse menu that accomplishes this:
As for a way to completely remove the global search box: unfortunately, this is not possible without some development.
One easy thing you could do via the user interface would be to change the display text that appears in the global search box, so that it clarifies that this will search across all institutions by default. Where the text in light grey shown in the global search box reads "Search" by default, you can customize this in Admin > Settings > User interface labels: change the setting for the globalSearch:
Fully removing the global search box will require code changes. The least intrusive / easiest way to maintain this through future upgrades would likely be to try to make the change in a custom theme plugin, so you don't have to alter AtoM's core code directly. We have some resources on developing custom theme plugins, such as: