For those of you, like me who wondered why the Pro-Tem instead of the Senate Majority Leader is closer in line for the presidency(like me)...
I found this at Quora, and have cut/pasted it below.
From J. David Gillespie (Political Scientist and writer)
First, the Senate Majority Leader, though generally considered in practice the political leader of the Senate, is NOWHERE (not just not the third person) in the presidential line of succession, and that is because his position is not recognized in the Constitution.
Secondly, the two people who are recognized as the third and fourth (after the Vice President) in that line of succession are members of the House and Senate respectively, holding the constitutionally-recognized titles of Speaker of the House and President Pro Tempore of the Senate. The Speaker is the parliamentary leader of the House in the governmental sense, but she is also the holder of a constitutionally-recognized position. (See Article 1, Sec 2, especially this: “The House of Representatives shall chuse their Speaker….”). The Constitution also recognizes the Senate President Pro tempore (see Article 1, Sec. 3), though in practice the President Pro tempore lacks the governmental power of the Speaker and is now (and for more than a century past has been) the holder of a position awarded as an honor to the most senior member of the majority party in the Senate.
Thirdly, the constitutional line of succession does pay deference to the primary or upper position of the Senate by virtue of placing the Vice President as second in line of succession. Although the Vice President is not a member of the Senate, she does hold the constitutionally-established title of president of the Senate, the framers apparently having had the intention of the Vice President being the actual presiding officer of the Senate, with President Pro Tempore serving in times of absence from the Senate by the Vice President. In practice, the Vice President is rarely in attendance at the Senate and most of the VP’s routines are those of an executive, and it is usually a Senator other than the Vice President or the President Pro tempore who holds the gavel and presides. But the Vice President does have one legislative power and that is to cast the deciding vote in breaking ties, a responsibility particularly important for the first female VIce President, Kamala Harris, given that the party line-up in the current Senate is a 50–50 Democratic-Republican split.