@simpsoap
Edit: I basically repeated everything already said above, but maybe the OP will read one of them??
A small but constant leak as already discussed can dry up (the water component) leaving behind the reddish residue, and not have anything show up
on the ground. The leak could be from a leaking hose or hose connection, or a leaking radiator; either a crack in the top tank or even a failed O-ring (gasket) around the inside edge of the top plastic portion (tank) of the radiator.
If you haven't yet, I would scrub the red sludge off from the top of the radiator using a stiff brush, old toothbrush,etc, wash it off well, then dry it off.
As a test you could go ahead and replace the upper radiator hose and the clamps now,
using Toyota parts.
With a new OEM hose, new OEM hose clamps and cleaned off radiator check the top of the radiator and hose connection just a few minutes after starting a cold engine. ie: get the engine up to operating temperature in your driveway idling, shut the engine off after say 20 minutes, look for a leak. Come back after 15 minutes and look again during the day. Already mentioned but you can add a DYE INTO RADIATOR which will glow under UV light, that will help finding a small leak.
Also already discussed but you could also buy (or rent at an Autoparts store) a cooling pressure tester and pump up the pressure in the cooling system yourself and look for the leak.
Amazing deals on this Radiator Pressure Tester Kit at Harbor Freight. Quality tools & low prices.
www.harborfreight.com
Do not wait a week or even a day of driving before you look for the leak, it should show up in minutes after the system gets hot and pressurized and/or after the engine is shut down.
If/when you replace the radiator you could replace the Thermostat (OEM part) if you don't know it's age.