Well, one way to determine if the head gasket is leaking compression/combustion into the cooling system is to connect a common radiator pressure tester to the radiator filler port before starting the engine when cold.
DO NOT pump any pressure into the radiator with the tester pump, you are only going to use the pressure gauge to see how soon, and how much pressure, registers on the pressure gauge.
If the pressure gauge shows a rise of pressure up to or near the pressure rating of the cap within a minute or so after starting, that indicates the system is being pressurized by something other than normal thermal expansion.

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.

I have a 2305 with the same problem. It overheats every time I cut the grass. My daughter and her boyfriend were using it and really overheated it!!! Now the head is cracked and the block is scored. The JD dealership is telling me a price of $14000.00 to rebuild it. I have read of this overheating problem in a lot of places but no one at JD has heard of it!!! We have to figure this out!!! I found a brand new engine in Illinois for around $4500.00 but don't have the cash right now and live in Alberta, Canada.

I am having the exact same problem. First I notice I have to constantly clean the screen in front of the radiator and the grill in front of that. Lately I've been overheating after 30 min or so of heavy mowing. I think I've blown the head gasket this time. It seems like the radiator is just to small to be efficient for the this tractor. Does anybody have a solution?

I have the same problem with the 2020 2025R tractor we purchased a couple of months ago. I have to carry a blower around so when it cuts off I can blow the radiator off. John Deere should fix this problem! A tractor is only worth its ability to perform.

MIne us a 2012 I believe. It has 320 hours currently. At 110 hours a full rebuild was needed and performed by Bader & Sons in MIchigan. They are the Deere dealer here. The screen in front needs cleaned every 15 minutes when mowing and overheating was still a big problem. The rebuild cost $2954.00 and my tractor is currently tore down again with a blown head gasket and cracked cylinder head. Mine is going to be traded in for a Kubota in the spring. Enough is enough.