How are through-hole boards mass-produced?

Mass production THT parts are either done with cheap people, or with axial insertion machines. You can find lots of videos on youtube:

https://www.youtube.com/watch?v=CCBST-qk2t8

https://www.youtube.com/watch?v=P17J5RzbsgQ

They pull the part off the bandolier, cut the leads, bend the leads, insert the part, and bend the leads on the opposite side in one stroke.

There are through hole assembly machines!
https://www.youtube.com/watch?v=2sm_qdeconI

But not all parts are rigid enough to be handled this way. Most of the through hole parts are assembled by hand before wave solder. Or also soldered by hand. This depends on other conditions, like dual loaded boards, or quality requirements.

Some through hole components can be placed with regular pnp. These are most found in short pin connectors. You can even reflow solder these by using the pin-in-paste method.

Most through hole parts remaining today are high power silicons, LEDs, electrolytic capacitors and connectors. Those will never die.