Why is it that March 21 is so much colder than September 21 despite the length of daylight, and thus the sun's angles, being the same? Why isn't the time around June 21 the hottest time of year with the sun's angle the highest and the most amount of daylight?

The warmest season of the year follows the solstice because the air and ground have cooled over the winter. It takes a while for the higher spring solar energy input to warm things back up.

A similar lag occurs as solar energy drops in the fall and it takes a few weeks for the summer heat to radiate into space.

Imagine the warmth is a big tank of water with a leak in the bottom. The leak represents heat radiating into space. After the slower supply of water representing winter, the level in the tank is low. Only past the equinox does the water input increase enough to overtake the leakage and start building the level back up again. After the solstice the water supply starts falling off again, but it's still faster than the leak and it takes a few weeks until the supply rate falls back to the same quantity as the leak. That's when the water level is at its highest.

Of course with the Earth's warmth there's also periods when heat radiation into space is impeded by clouds or heat is redistributed by winds and ocean currents.