I've seen this work on some Honda's that are old as all hell and overheat WITH a radiator for some reason, BUT I probably wouldn't recommend it to anybody. Without the Thermostat, the coolant will flow through the engine constantly and not allow it to warm. This would give an effect similar to an air cooled engine such as an old VW Beetle. The engine is running cool and cause the car to sludge up bad. In the long run, it's not worth it for a cheap part like a thermostat and even if that not the only problem, it's still much better than what you have to deal with in the long run. Horrible gas mileage caused by sludge. Sticking valves, and probably having to have the engine cleaned or replaced in the future.
Edit: If I were you, I'd offer to do the job myself if it's just a problem. An easy 50 bucks, which is wayy cheaper than they'd pay at a shop. Drain the system for them, put some new fluid in, replace the t-stat and walk away with gas money for the week. Haha :hihi: