No. of Recommendations: 3
As for the accountability of a new fourth branch, why couldn't that be solved by independent committees/commissions appointed by Congress and perhaps also with new members put up for a general popular vote? Heck, just put all the members on the ballot every 4 years like the president. And also impose term limits on the members of those independent groups.
What problem are you trying to solve, then?
Are you just creating a "second President," giving having that second President be named by the members of a popularly elected commission (ie. a second Electoral College), and giving that second President control of some subset of the cabinet departments - like Justice and Treasury? That's a recipe for disaster, because you need those departments to function with the other departments of government. And you're in no better of a position. You still need someone to be named to actually run these agencies - if it's a second President, or some "commission" that exists just to name people to run these departments, if the commission is a political body then the folks they appoint will be named politically as well.
There's evidence to support independent agencies. Many countries have their central banks somewhat independent of the elected government, and obviously the judiciary is independent in many countries as well. But the trade-off to that independence is that the agencies don't have to be responsive to the electorate....and that's not necessarily a good thing for agencies like Justice or Treasury.
When considering independent agencies that are outside of normal political pressures, you can fall into the trap of imagining their benefits when headed by a Solomonic leader who will make good decisions. Wouldn't it be great if the Attorney General wasn't beholden to the President, and DOJ would therefore be able to do only good things? But you also have to think about how that independence works when those agencies are led by truly atrocious people, or people with their own political agendas. Imagine what happens with ICE in Minnesota or Chicago if Kristy Noem is independent - not subject to being removed when the President and/or Congress start facing political costs from how the Department is being run? Or Pam Bondi running DOJ without having to care about any blowback from any choices she makes?