Ask an astronomer
"I thought the reason why we had to add the extra second is because the Earth's rotation is slowing (i.e. the days are getting longer)"
That's correct, up to a point. When the length of the second was standardised in the early 1970s (as part of the SI system of units) in terms of the oscillations of a caesium atom, the value was chosen to match the previous international time standard called Ephemeris Time, which was based on the orbital motion of the Moon and planets.
Unfortunately, that timescale had been defined in such a way that 86,400 seconds were a very good approximation to the length of the day circa the the early 19th century. The Earth's rotation had slowed somewhat in the intervening 150 years, so a day in 1970 was several milliseconds longer than a day in 1820. Those milliseconds add up over the course of a year to give an excess of a whole second, and hence the need for a leap-second every so often.
"But actually making every second longer ... would seem very unobtrusive, if managed right."
High-precision timekeeping now pervades our lives to such an extent that it would be utterly impractical to adjust the length of the SI second.
In any case, the Earth's rotation will continue to slow down, so even if we re-defined the SI second to match 1/86,400 of the current length of the day, we would only be making trouble for the future.