Why, physically speaking, is the tropical year almost but not quite as long as the sidereal year?
The seasons are caused by the orientation of the Earth's tilted axis relative to the sun. E.g the north pole is tilted towards the sun during summer of the northern hemisphere, and away from the sun in winter.
Now imagine the Earth being magically held in place, instead of orbiting around the sun. This would almost stop the seasons from advancing, but not entirely. There would still be a slow 26000 year cycle of the seasons, because the Earth's axis precesses much like a spinning top. It's this effect that speeds up the cycle of the seasons by about 20 minutes each year. (1 year/26000 is about 20 minutes)
Is there a reason it precesses at that particular speed, some resonance or tidal locking or something? Or is it basically a random initial condition?
It's pretty random. It's mostly caused by gravitation from the sun and moon, and counteracted by the Earth's spin; faster spin would mean slower precession.
The solution (adding a day every four years) doesn't seem specific to either.
Circling back to the starting point would be a sidereal year, which isn't what we want either.
What the calendar is trying to synchronize to is not the sidereal year, but the tropical year: the equinoxes and solstices should occur around the same date each year, and the seasons shouldn't shift relative to the calendar over the years. Because of the precession of the earth's axis, this tropical year is about 20 minutes shorter than the sidereal year.