The thermostat is not a mandatory component, as when it is cold, it stays closed to warm up the engine faster; and when gets warm it opens up, allows the engine to stay cool. Without the thermostat you will have unrestricted flow to cool the engine, and the only downside is that it will take longer to warm up the engine. In a car, it would take longer for the heater to work, if it is cold out, but that is not a problem in a boat.
If the housing is too rusted, the option is to go without the thermostat, or you may be able to find a new housing in E-Bay, but it doesn't look too bad from your pictures.