I'd go further and say that the rest of the cabinet should also be elected from the House (not just the President). This ensures a cabinet member is answerable to both the House and to an electorate. Though the two-year terms in the US House are probably too frequent for a parliamentary style system like this (perpetual campaigning).

The norm in a parliamentary system, is the Prime Minister is chosen by the House, and then they (effectively) appoint the other government ministers. If the house doesn’t like their choices, they can always sack the PM. A political party is allowed to adopt internal rules constraining the PM’s freedom of choice in appointing ministers, but the constitution itself doesn’t.

I think for the US, a South African-style system in which the offices of President and Prime Minister are combined, would be more comprehensible to the American public than a (mostly) figurehead President combined with a Prime Minister, as in Germany, India, Ireland, Israel, Malta—the essence of the parliamentary system is subordination of the executive to the legislature, splitting the head of state and government is a complication which adds less value and would be a bigger change for Americans to comprehend.

Australia’s federal constitution says a serving federal minister must be a serving member of the federal Parliament-either the House or the Senate. However, there is a rarely used exception—you can be appointed a minister without sitting in Parliament, but only for a maximum of three months. Part of the intention behind the clause is an Andy Burnham-style scenario in which some outsider is chosen to be PM (or more rarely another minister), and then they can start serving in that role while running in a by-election (special election).

This was actually used in 1968 when John Gorton was chosen to succeed Harold Holt as PM, after Holt’s disappearance at sea-Gorton was a Senator, and while it isn’t strictly speaking unconstitutional for a Senator to be PM, by convention they must be a member of the House-so Gorton became PM as a Senator, resigned from the Senate, Gorton then ran for Holt’s House seat and won, becoming a member of the House-all while remaining PM the whole time-so, in the period between his resignation from the Senate and his election to the House, he spent 24 days as PM without being either a member of the House or Senate, but that was under the three month limit. He also spent around 14 days serving as PM from the Senate-which is technically constitutionally allowed, but nobody wants that in practice except as a temporary measure.

The original purpose of the three month limit - Australia’s first ministry was sworn in on 1 January 1901, but the first Parliament wasn’t elected until March. Edmund Barton was chosen to be PM by Lord Hopetoun, whom London had appointed as the first Governor-General; Hopetoun’s first choice was Sir William Lyne, the Premier of NSW; but that choice was very unpopular with both the Australian and British establishments, in response to which Lyne ended up declining the role and recommending Hopetoun appoint Barton instead, which recommendation Hopetoun accepted.