I recommend to read the section about data integration and the usage of the thus normalized data by the Bioconductor folks: https://osca.bioconductor.org/integrating-datasets.html#using-corrected-values

In brief, data integration is mostly important for dimensionality reduction and cluster identification where you want to be able to identify cells that are very similar to each other. I.e. you want to avoid that neurons in sample A are sorted into a different cluster from the same type of neurons from sample B due to batch effect. To this end, you really want to zoom into those features (genes, eigenvectors, ...) that capture the essence of a cell's biology, i.e. you will probably reduce the final data set that is used for UMAP and clustering down to 20-30 principal components that are usually sufficient to align the cells across technically distinct samples.

As soon as you're going to the gene level, though, it is better to work with the raw data, especially if you happen to have replicated samples, which allows you to apply the typical bulk RNA-seq tests while accounting for the batch effects as covariates. If you want to draw conclusions on a single-gene level, you need to go back to the actual read counts per gene and apply the proper tests on those.