If the mills ratio of a Heckman selection model (with/without exclusion restriction) is not significant, shall I prefer to estimate my model with OLS instead? Or is it better to use the estimates from the heckman model although there seems to be no sample selection?
In this case you would have no selection problem, so you could use OLS on the non-censored observations. This can be tested by what you did. So you are correct, see e.g. also wikipedia
"so testing the null that the coefficient on $\lambda$ is zero is equivalent to testing for sample selectivity."