I assume you will already get balanced treatment assignment within the randomization strata. Why would we still gain improvement in precision by adjusting for these factors in the model? Does not make sense to me intuitively.
Per a ScienceDirect source, pertinent comments:
Stratified randomization prevents imbalance between treatment groups for known factors that influence prognosis or treatment responsiveness. As a result, stratification may prevent type I error and improve power for small trials (<400 patients), but only when the stratification factors have a large effect on prognosis.
So, apparently, it is advisable to mitigate a significant factor that can impact Type I Error.
[EDIT] With respect to the model itself, the precise answer relating to model improvement, is that you have now separated out more clearly the expected mean barring contribution from a known factor that influences 'prognosis or treatment responsiveness'. Also, the variance of the underlying mean is more precisely identified. The latter estimates may be of use in say a simulation exercise with possibly many such known influencing factors included (or excluded).