i’m on Android 13 also (vanilla LOS [ungoogled]) and it’s the first version i’m running unrooted. i did once change webview to Bromite’s, i think 2 or 3 Android versions ago, but on the next one i could not manage to change it again, so i’m stuck with the default
i’m glad these custom tabs are a thing now tho, since all the apps that i’m using that need some webbing seem to be ‘powered by Mull’ now