Best common metric for comparing classic time series forecasting methods (ARIMA/Prophet) with ML approach?
In general the most common metrics mentioned in several articles (e.g. like this one) are the same we also commonly use for non-time series prediction:

MAE Mean absolute error
MSE Mean squared error
RMSE Root mean squared error

Outside of linear regressions I have not seen R² used that often to validate prediction models. In fact it isn't even one of the out-of-the-box metrics of xgboost for interval scale prediction tasks. So you should be fine using RMSE for everything.
Edit:
Prophet at least in the R package covers RMSE, MAE and MAPE so you should be fine.