As the phrase “stronger” can be interpreted in different ways, this is a bit of a challenge to answer. While it may be harder to achieve a statistically significant result using random effects models, the best answer is probably, “No...as longs as the model is specified correct.” That is to say, if the model with mixed effects best describes the population, then it will be the stronger model...vice a versa if it were the fixed effects model.
For the second question, again, there is ambiguity in what is meant by “proper” in this context. First, if you Ramsey RESET test fails to indicate linearity, this should be interpreted as any other model assumption testing protocol: there is not enough statistical evidence to reject the claim the the model is linear, but it still does not prove that it is linear. Regarding the second part of the second question...the fix for failing the RESET test is to include higher order terms...not to include the random coefficients (unless I really missed something, in which case, corrections are definitely welcome). I'm led to believe that inclusion of random effects instead of higher order terms could result in statistically significant findings for the random model (when in truth, there is no random effect).