Another way to test for compression/combustion leakage into the cooling system is very "old school" but still valid, however, it requires caution and a stop watch or timer.
For this test, begin with a cold engine. Remove the fan belt from the pulleys and get the belt away from the crank pulley. Removing the belt disables the water pump and this is necessary because the swirl of the pump could spoil the "quiet stillness" needed to observe the escaping gases (if they are present).
Remove the radiator cap. Top off the radiator with coolant so that there is no empty space below the inner radiator cap seat flange. Start the engine and your timer (or note the time on your watch). DO NOT RUN THE ENGINE MORE THAN 2 MINUTES without a fan belt.
With the engine at idle RPM, watch the coolant in the radiator filler opening.
If you see what look like "air bubbles" begin to burb out of the radiator opening within a few seconds after starting the cold engine, that is "compression/combustion gases" escaping from the cooling system.
Be advised that on some engines and depending on how bad the compression/combustion leak actually is..........what you might see instead of a few bubbles............could be a geyser of coolant shooting up from the radiator filler opening, so don't stick your face too close to the opening, or directly above it, and you would be wise to wear some kind of approved eye safety goggles or face shield.
If you see gas bubbles or a geyser within 2 minutes, you definitely have compression/combustion leaking into the cooling system.

Now, about the tendency for the "grass screen" to clog quickly.
That is unusual unless there are some other circumstances or conditions that are promoting a "cloud of debris" in front of the radiator while you are mowing.
Have you looked really close at what the screen is catching?
How tall is the grass or weeds (as I don't know what you normally are cutting) that you are mowing? Does the grass or weeds usually have "seed heads" or dandelion fluff?
What about other types of plants such as milkweed (that also have airborne seeds connected to "fluff parasails")?

If the grass/weeds you are mowing is fairly tall, has seed heads (or both) and you are mowing with the front bucket down low enough to "skim" the plants.......when the plants rebound after the bucket passes them, they are flinging fluff and seeds into the radiator air inflow zone where the screen can capture them.

If that scenario does not match what you are doing, then you need to look closely at the exact circumstances that ARE taking place when you mow.
Something is causing the tractor to draw debris filled air into the radiator and something is causing that debris cloud to be present.

If you are cutting at a higher level in sparsely vegetated turf, it's likely that all this "chaff" is escaping all around the edges of the deck instead of being neatly discharged out the directional discharge chute.
If that is what is happening, you (or someone) might need to engineer some sort of skirt in front of the deck to stop the chaff and clippings from escaping toward the front of the tractor.