Yep, this is doable, but the way NetSuite behaves can feel a little unintuitive until you separate “who can run it” from “who can edit/save it.”
How NetSuite actually controls it
A saved search has two main access layers:
Audience / Available To (who can see and run it)

Edit / Save rights (who can change the definition of the search)

If you add people under the Employees list (or set an audience that includes them), that controls visibility. It does not automatically mean everyone can edit.
Best practice setup for what you want
Step 1: Let everyone run the search
Set Audience = Public or restrict to the roles/groups that should be able to run it.

That’s your view-only crowd.

Step 2: Limit who can edit/save
Use Owner and “lock down the edit path”:
Set the Owner of the search to a single admin/power user (or a generic “Reporting Admin” employee).

Do not share it as a “work-in-progress” personal search owned by a regular user.

Then make sure those “view-only” users don’t have permissions that allow them to edit searches globally.
In practice:
Users who can run the search will see it and run it.

Only users with the right permission (usually a high permission like Saved Search or similar “manage searches” capability) will be able to edit the original.

The part you’re worried about: “Save” vs “Save As”
Even if someone can’t edit the original, NetSuite commonly allows:
Save As (copy it and make their own version)

That’s normally fine and actually desirable, because it prevents overwriting the master search. The risk is only if your role permissions let them do actual “edit” on saved searches.
So does the Employee list restrict the Save button?
The Employees list is for who can access/run the search. The ability to Save / Edit is still governed by:
who owns the search

the user’s role permissions related to saved searches / lists / reports

Meaning: checking “edit” related options at the role level can absolutely give broader edit capability than you intend.
A safer pattern (what most teams do)
Create one “gold” search: AP – Open Bills (MASTER)

Owner: Admin or Reporting Admin

Audience: everyone who should run it

For your power users: tell them to use Save As and create AP – Open Bills (Jane) variants

That way nobody “accidentally overwrites” the original, and you still get flexibility.

Quick check question (to be sure you don’t get surprised)
When your regular users open the saved search, do they currently see Edit at all, or only Customize / Save As?

Because the answer depends on whether their roles include saved search management permissions. If they don’t, you’re already safe: audience controls visibility, not edit.