privateWindowHomepage.uc.js

// ==UserScript==
// @name           Private Window Homepage
// @version        1.0
// @author         aminomancer
// @homepage       https://github.com/aminomancer/uc.css.js
// @description    By default, private windows are opened to about:privatebrowsing, regardless of your homepage or new tab page preferences. Once the window is opened, opening a new tab goes to your new tab page, and pressing the home button goes to your actual home page. But the first tab of a private window is always opened to about:privatebrowsing. This behavior is coded right into OpenBrowserWindow() but we can change it. This script simply removes the part of the function that manually sets the URL to about:privatebrowsing. So private windows will now behave like ordinary windows in this (and only this) respect.
// ==/UserScript==

(function () {
    function init() {
        window.og_OpenBrowserWindow = OpenBrowserWindow;
        eval(
            `OpenBrowserWindow = ` +
                OpenBrowserWindow.toSource().replace(
                    /\N*\s*if \(\!PrivateBrowsingUtils\.permanentPrivateBrowsing\) {\s*.*\s*defaultArgs \= \"about\:privatebrowsing\"\;\s*\}/gm,
                    ``
                )
        );
    }
    if (gBrowserInit.delayedStartupFinished) init();
    else {
        let delayedListener = (subject, topic) => {
            if (topic == "browser-delayed-startup-finished" && subject == window) {
                Services.obs.removeObserver(delayedListener, topic);
                init();
            }
        };
        Services.obs.addObserver(delayedListener, "browser-delayed-startup-finished");
    }
})();