(prescript: I don't know how any of this is laid out on iOS, so I'll just speak on the Android layout)
Hi! I think the mobile navigation bar could use some reorganization.
It would be nice if instead of having a home button, the home screen were accessed with a swipe-up gesture from the navbar (or swipe-down, if it's situated on top). This would free up space on those narrow screens for something I think is much needed: a tab-close button.
If a close button is added, you could replace the "close tab" item in the tabs button's press-and-hold menu with "Undo close tab / Reopen closed tab." On the home page, the close button would be irrelevant so you could put the private browsing toggle there.
A fun bonus is that this would result in more layout consistency with the custom-tab window—'X' button on the left.
I also think a gesture will make the "home page" concept more coherent. It's always been a bit unintuitive how it's not a 'tab' like it is on desktop. I used to be afraid of that button because I worried it would close my current tab or create a new one and over-crowd my tab list. Which is a bummer! It does neither of those things, but it took me a while to realize that.
A gesture that pushes the webpage out of view to reveal the homepage would more clearly communicate that concept, it would just need to not be animated too slow and respect the device's reduced-motion preferences.
I know the left-right switch tab gesture has a configuration option. If this gesture were optional, disabling it would bring back the home button and might forego the close button on non-tablet devices.