The proper equation to use for that is n(n+1)/2 =x which is the solution of 1+2+3+..+n = x
To get 'n' from x, note that for big numbers n * (n+1) is very close to n2. So you can just take the 12,614,400, double that (to account for the division by two in the above equation), then take the square root and round down, and that's how many items the full song will have. So a 1-year song could have 5022 items. Call that 5000 items.
(5000 * 5001)/2= 12,502,500 sung items in a year, which work out at almost 24 sung items per minute, which is very close to the original estimate. So you could start singing the "5000 days of Christmas" at midnight New Years Eve and complete the whole song by midnight New Years Eve the next year, ready to start again.
However, if you made a version of the song with N = 12,500,000 items, and it took a year to sing just the last verse, then to sing the whole song would take N(N+1)/2 steps, which would work out at 6.25 million years to complete the whole song. Which makes perfect sense. You're singing every version of the song from length 1 to length N, so the average length of a verse is N/2, and you're doing that N times. So you're singing 12.5 million verses but the average length of a verse is half the longest, so with 12.5 million items taking a year, it takes half that or 6.25 million years to sing.