Any brown buildup is usually indicative of rust within a system because the coolant hasn’t been changed as required. And rust residue for your alloy Ecotec block equates to the cast iron cylinder liners (if wiki can be trusted) corroding.
So, it seems you need to consider flushing the system clear any of the brown residue to restore cooling efficiency. This usually means a chemical flush. BUT if an engine cooling system hasn’t been well maintained, such a flush could result in a number of new leaks developing and it depends how bad things actually are. Usually post chemical flush leaks occur within HVAC radiator behind dash, engine radiator and/or possibly hoses, etc...
So I’d suggest pop by a decent mechanic you trust as they can test the radiator cap, the radiator top where the cap sits as well and the general system for coolant leak. They should also test the radiator water for combustion gasses. This should be a five to ten minute job and they shouldn’t charge much for this service but it will give you valuable information as to how big a problem you may have. They could then give you a price to service the cooling system should you desire.
But, if it’s found it’s just the cap, you can do a clean water flush (with low pressure compressed air to agitate the water) and you should be able to get most of the crap out. Remember to have the heater on during the flush. Fill up with new “coolant” and pop on a new cap and hopefully all is good from then on.
Be warned though, driving a car around which does not build up water pressure will lead to loss of water to the point the temp gauge wont work in some cars. This can and does killing engines. So don’t put it off and if you must do short trips, check water level before each drive.
WARNING: on an engine, always check that the hose has no pressure by squeezing the radiator hose before opening the cap to its first stop. Wait for any pressure to be released, if any, then remove cap. If the hose is hard when squeezing, the cooling system has pressure and you should not take the radiator cap off until the coolant pressure drops as it cools. Taking the cap off a hot system can result in serious burns.