Major Plot Hole on the TV Series "Designated Survivor" (spoilers if you haven't watched the show; no spoilers if you're a regular)

I love the TV show Designated Survivor, which depicts a low-level member of a presidential cabinet thrust into the Oval Office when a terrorist attack during a presidential State of the Union address wiped out everybody ahead of him, along with almost all of Congress, and the Supreme Court. While the show is accurate in its portrayal of having someone deliberately stay away from a large gathering of government officials (designated survivors were used during the recent inauguration as well as when President Trump made his address to Congress a few weeks later), part of the story-telling either ignores or overlooks some facts in how a presidential vacancy would actually be filled in such an emergency. Being the total nerd that I am, I feel compelled to point them out.
There are three major places where presidential succession is covered: in Article II, Section 1, Clause 6 of the Constitution: In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected.
The 25th Amendment clarified the first part of that, up to where it says "...and the Congress may by law....": In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.
Additionally, the Presidential Succession Act of 1947 is the most-recent legislation in which Congress covered what happens if there is neither a President nor a Vice President in office. On top of that, the second section of the 25th Amendment covers a Vice Presidential vacancy (prior to this amendment, the vice presidency was always left vacant until the next election if it were to become vacant due to the death, resignation, removal from office [which has never happened], or ascension to the presidency of a sitting Vice President): Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress.
Now, in the show (here's the spoilers if you haven't watched the show), what happened was, after the attack, HUD Secretary Tom Kirkman, who was serving as designated survivor, was sworn in as President. Later on, it was discovered that two members of the House of Representatives also survived the attack; one of them became the new Speaker of the House, and the other was appointed Vice President. Immediately after the new VP was sworn in, an assassination attempt was made against now-President Kirkman, in hopes of the new VP (who the terrorists wanted in the White House all along) then succeeding to the presidency.
It is in the way succession works in real life that the writers of the show either took dramatic license, or else screwed up. Primarily, notice how the clause of the Constitution I cited lists that anybody other than a Vice President can only ACT as President, not actually BECOME President. Additionally, according to the terms of the Succession Act of 1947, if a person higher up the list than the person who is currently acting as President becomes available, that person can then bump out the acting president, and assume the job themselves. Now, one can argue that an acting president in this kind of situation would be treated as actually holding the office (being addressed as "Mr. President" as opposed to "Mr. Acting President" or by their previous title), but the convoluted process the writers set up to try to get the congressional stooge into office would not be necessary in real life.
In order for someone to become Speaker of the House, they need a majority vote of the House...that is, of the members alive and in office at the time, NOT the total number of seats (435). In the story, President Kirkman asked Congressman MacLeash (the guy the terrorists wanted to be president), to become Speaker, and he declined, hoping to be appointed VP as per the terrorists' plans. But, in real life, he, who at the time thought he was the only living member of the House, as the survival of Rep. Hookstratten was not yet known, could have theoretically elected himself Speaker of the House. After having elected himself Speaker of the House, he could have exercised his rights under the Succession Act of 1947, and bumped ACTING President Kirkman out of office, claiming it for himself. The Act was never amended to address what happens when and if an acting president under this act were to nominate a new vice president under the terms of the 25th Amendment, so one can infer that the acting president remains acting president, while the newly-confirmed vice president would remain vice president.
So, there's not a whole lot of story to be told if the writers had followed how the laws actually work, but they're actually making a major error in how things would work in "real life." Yes, this is something that would have made my late grandmother ask me "Can't you just shut up and watch the program?" but as anybody who knows me knows, I like to nitpick!