Does gridsearch on random forest make sense?

You are right that randomness will play a role (like with many other algorithms including MCMC samplers for Bayesian models, XGBoost, LightGBM, neural networks etc.) in the results. The obvious way to minimize randomness in the results of any hyper-parameter optimization method for RF (whether it's random grid-search, grid search or some Bayesian hyperparameter optimization method) is to increase the number of trees (which reduces the randomness in the model behavior - albeit at the cost of an increased training time). Alternatively, you construct a surrogate model on top of the results that takes into account that the signal, of where the best model in the hyperparameter landscape is, is noisy through an appropriate amount of smoothing/regularization.