No to getting rid of the thermostat. Among other things, it could affect passing emissions test. If you replace, before you take the old one out note the position - it is easy to install upside down. As mentioned, temp gauge is not necessarily reflective of true engine temp. Other cooling system issues are clogged radiator core, old coolant, wrong mixture of water/coolant. Coolant is hazardous waste so be sure to capture old coolant if you drain.