I have an absolutely ancient Motorola Droid X2 (released 2011) serving as an always-on weather display. The stock browser finally started choking on the weather site because it doesn't have up to date root certificates, and can't by made to bypass/ignore this. This device is way past the point of having the app store work, but we can get a decent Firefox on there, and while it too will complain about certificates, it can be made to install a permanent exception.

1. ADB: Get ADB. You can download the portable version here.
2. Debug mode: Turn on "USB Debugging". On my device, this was under Settings → Applications → Development.
3. Test ADB: Run "adb devices". The device may ask you to authorize. If successful, you should see "List of devices attached", "502…207 device"
4. Find an acceptable Firefox (or whatever app): Get an APK for the latest Firefox to support your android version (which can be found in Settings → About). This site shows all variants of Firefox, and can be made to search for other APKs, including with an Android version filter. For my Android 2.3.5 device, Firefox 47 was the pick.
5. Install it: Down the chosen APK(s), and for each, just run "adb install <your_apk>".

Note: don't navigate to random sites, as this old browser is likely to have security vulnerabilities. That said, if you're using these directions, your device is lightyears out of support anyway.