The petition has been around for so long, it doesn't really seem like it's going anywhere unfortunately, is why recently I've been saying to complain/leave feedback to Nintendo directly.
As we know, the browser is already built-in and there, it's just hard to use– which is the frustrating part! The official name of the browser is Netfront and it's made by a company called Access which presumably has a partnership with Nintendo and other companies. They also made the Wii U and 3DS browsers. So Nintendo is already paying real money to Access for the browser, only to ultimately lock it down!
For bdns/switchbru, if they introduce a real browser, we would continue to just operate as normal websites, and likely be able to be bookmarked within the Switch browser. There were a lot of for-DS, for-Wii, etc sites like this, so It would be a win-win!