A screen over the intake would cause too much of an air flow restriction.....not to mention disruption of the aerodynamics of the aircraft.
The facility where I used to work overhauled and tested turbine engines.
During testing a screen was used to mitigate foreign objects from being ingested....if Thatcher to happen it would mean many hours teardown, determining what parts were damaged and then replacing them.....all that would of course increase the cost.
Sure enough, on several occasions, a small piece of the screen assemby would dislodge and be immediately ingested.
When that happened it was always a very big and expensive deal.
What parts of your test screen got loose?