PS4 1.76 Permanent Internet Browser Mod Payload, and recently PlayStation 4 modder MODDED WARFARE shared a video tutorial on enabling the PS4 1.76 Internet Browser without signing into PSN.
The guide uses the PS4 WebKit Playground, which can also be accessed in several flavors courtesy of e✘treme, and to quote:
How To Enable The Internet Browser on a PS4 Without PSN
How to enable the Internet browser on a PS4 with firmware 1.76 while not signed into PSN. With the Internet browser enabled you no longer need to use the user guide method to access the web kit playground.
PS4 Browser Patch: WebBrowserPatch.bin
PS4 Debug Settings Injector: ps4debugnetcat.zip
Thanks once again to @HydrogenNGU on Twitter for the heads-up of this video tutorial.