Really? The Secretary of State? That sounds a bit… odd, especially since I thought the line of succession was the same whether the President died or was removed from office.
It is the same.
And considering that it's an unelected office (but then again, so are most), I could see many people objecting; John Adams, for one, would be turning in his grave.
Hardly; Secretary of State was commonly a springboard to the Presidency.
Jefferson, Madison, Monroe, JQ Adams, and Van Buren were all SoS and then President; five of the first 10 SoSs and five of the first eight Presidents.
Other early SoSs included former and future Presidential candidate Henry Clay, former Vice President Calhoun, future President James Buchanan, and former Presidential candidate Lewis Cass.
I knew the latter bit; however, I remembered that upon Harrison's death, there were some suggesting that Tyler should become Acting President and only serve until a special election could be held.
The original succession law provided for a new election in the case of double vacancy, with the president pro tem as "acting President". It was changed in 1886, with succession going to Cabinet members only and the provision for new elections dropped. In 1947 the law was revised again to the present status: Speaker, president p.t., then the Cabinet.
One little known factoid. At the time of the 1916 election, U.S. foreign affairs were in a state of on-going crisis, with the imminent possibility of war. It occurred to incumbent President Wilson that if he were defeated for re-election (as nearly happened), the resulting three months of "lame-duck" government could be ruinous. He therefore decided that if he was defeated, he would appoint the Republican President-elect (Charles Evans Hughes) as Secretary of State, and then he and Vice President Marshall would both resign. Hughes would become President immediately, avoiding that dangerous delay.