How to train and evaluate machine learning models with growing/changing datasets over time
The solution you described is definitely an option.
The other motivation for retraining is that the distribution of new data has drifted away from that of the original dataset and that warrants retraining to maintain the same level of model performance.
In such situations, It is possible that you may actually throw away some of your older training data because they may have become stale and include a part of newly available data for training. The remainder new data can then be used for out of sample testing and validation.
You also want to be able to figure out when to trigger the retraining. This is usually done by an independent drift detection process, which keeps track of some metric related to your Business and triggers retraining if it dips below a predefined threshold.