Did you search for Hyundai Sonata forums that may have members knowledgeable for help?
When you removed the cylinder head, were you able to see a damaged head gasket? There are reputable and unscrupulous dealers and repair shops that will say anything to unsuspecting owners and proceed to pad repair bills for repairs that weren't needed.
Overheating may be as simple as air in the cooling system requiring correct bleeding or as complicated as a clogged system never having periodic coolant flushing and new antifreeze/water at recommended intervals (mileage or years). Some thermostats wear out and stick in closed position leading to overheating while others stick open and run the engine colder than normal. Cooling fan(s) wear out or fail. Some cooling fans have more than one speed. When an engine begins to overheat, the coolant sensor, normally sends temperature signals to the engine computer all the time the engine runs. Signals let the ecm or bcm drive the temperature gauge needle and cycle the cooling fan on and off to prevent coolant from boiling over. A reader capable of displaying coolant temps would be helpful to you. Operating temps range from 175F-220F. Cooling fan(s) should blow hard and loud if its a single speed fan. Most vehicles should have a standard 87A relay for the cooling fan with either four or five pins. Shorting relay socket terminals (paper clip or small gauge wire) 30 and 87 should send 12v to power the cooling fan immediately regardless of ignition switch position. If you can do this, listen to the fan. A single speed fan should blow with force otherwise it may be worn out. Fan cooling is needed to force air flow thru two radiators - the ac condenser coil and radiator to be effective.