John Tyler, the first vice president to ascent to the presidency, established the precedent that an accidential chief executive indeed becomes president in name and title in 1841. That was an open question at the time as the exact wording of the constitution prior to the 25th Amendment was not exact and it was unclear whether the "office" or the "powers and duties" were devolved on the vice president. Interestingly John Quincy Adams, son of a founding father and congressman at the time, was opposed to Tyler's interpretation and said the founding fathers meant the vice president would just become acting president (though I've seen arguments that Adams just insisted this because he disliked Tyler). After Tyler himself insisted on being president and congress ultimately went along, it was a set precedent until the 25th Amendment cleared the issue in 1967. After Tyler, there were seven more presidents not completing their term, only Gerald Ford took over under the new rules (until today).
What if Harrison wasn't the first president to die in office? I wonder how such a situation would have been handled earlier. George Washington just barely outlived his presidency and wouldn't have survived a second term.
Things probably would've turned out differently.
I will say it was to the country's benefit the first time this occurred was not 1865. By 1865, we'd gone to the VP twice and so the precedent was well-established. Constitutional crisis in the middle of a war and you know the Republican supermajority Congress would've shriveled down the authority of Johnson as far as possible, which would've not served us well say 1945 with the Roosevelt-to-Truman transition.
