Your wild thinking may cost you a lot more money and hassle.
Find the coolant leak first.
If your losing it at that fast a rate it has to be seen at some point.
Pressurize the cooling system to find the leak.
The coolant just does not disappear into the air without a trace.
You have the hoses, the radiator, the heater and hoses, the line that runs through the block valley to the rear, and water pump shaft seal..
If you had a head gasket leak into a cylinder the motor would not run so well.
If it is going into the crank case you should see that evidence on the dip stick.
The radiator has rubber seals around the side tanks. They can be temperature sensitive and either seal with heat or open a leak with heat.
Look under the area for signs of a leak. Anti Freeze usually leaves a trail.
Shame to replace a motor for some lesser outside coolant leak.