Since your problem seems to have been, indeed, the thermostat, removing it works as the solution. The car should still work fine as you say, specially in temperate climate. You'll just need to wait in the mornings (or whenever your car has an extended period of time with the engine off) for it to warm up and reach a decent operating temperature.

Even though cars can run without a thermostat, I'd reccommend you to replace yours if you can afford to.

The thermostat is a part of the cooling system and as such, removing it it affects the whole... well, system. The flow pattern of the coolant is changed, so the expected load work on the water pump would differ from the real one, this could cause you problems in the future. Same with the pipes. Since the control system will think your engine is hot all the time, as you stated the cooling fan will work more often than it was originally designed for. There's also the issue with the fuel mixture, which can become uncontrolled if the control system depends on the thermostat, so you could have higher consumption and a bigger stress on the injectors. Because of the same reason, you could have a higher level of non-clean emissions which will affect escape valves and the catalitic converter.

If your car is fairly 'old' (no computer, carburator, etc) then the consequences might not be too worrying. Otherwise, they could cause you trouble. Not everything I said has to happen and even if it does it might not be catastrophic... but still.

Since the various systems on complex machines are often related, changing a component in one of the systems could have an effect in others. Sometimes this effects are negligible, sometimes they are not.