Madrid AP to Burgos

Hi AWard! Welcome to the forum! I wish you a pleasant and fruitful stay on the forum.

Alsa buses go to the bus station in Calle Miranda (the only one in Burgos) that it's close enough to the Cathedral (you can see the location on a map on the timetables area of the web of Alsa). I don't think Alsa's buses from Madrid make other stops in Burgos (if they made them, it wouldn't be at a bus station but at a bus stop).

I'm wondering if the source of confusion is at the Madrid end.

The website states there are 16 buses leaving from Avenida de America and 10 buses from T4 Airport.

What that means is some but not all of the buses that go to Burgos depart from Airport T4 while all of the buses either depart from or make a stop at Avenida de America. Which may mean you need to take EMT bus 200 from the airport to Intercambiador de Avenida de America or wait a while.

Intercambiador de Avenida de America is a transit hub for Madrid Metro, Madrid buses, and intercity buses like ALSA.