There are huge downsides to removing vast swatches of the Federal government (like the entire DOJ or Treasury) outside of political accountability. There's reasons why we would want those agencies to be subject to a person who is directly accountable to voters. We won't have a clear understanding of how well the existing checks and balances worked to limit Trump until after he's been out of office for a few years, but I'd be very leery about putting vast amounts of the government outside of normal politics.

First, sure, an awful president is ultimately accountable to the voters, but they only have a say every 4 years. That leaves 4 years of unfettered non-"high crimes and misdemeanors" that such a POTUS can inflict on the country, such as Trump is inflicting now on crumbling American body politic (not to mention his several arguably impeachable offenses). Plus, impeachment and conviction are impotent if Congress is cravenly subservient to a "strong man" president, such as it is now.

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.

...Ok, as I'm writing this, I supposed ideally we wouldn't WANT to have the members of such independent groups to have to be concerned with - and perhaps even tailor their decisions to maximizing their chances of winning - the next election. But I'm sure some compromise could be devised. He says optimistically...