How did Saddam successfully hang onto power for so long after the extremely costly invasions of Iran and Kuwait, the punishing sanctions regime, and being a Sunni in a Shia-dominated country? Was it the brutality, the relative secularism, or was there something more to it?

There are two valid answers to your question:

In history, it's almost always a little bit about everything. Almost nothing happens in a vacuum.
If you have enough guns & bullets, you can do whatever the F ya want.

One of President HW Bush's proudest diplomatic achievements was the multinational coalition he led to enforce UN demands: namely, Iraq must leave Kuwait. The list of coalition nations was astonishing, and included many Arab nations as well. (Israel wasn't part of it; Israel didn't even respond when Iraq attacked it with scud missiles, because America feared that Israel's involvement might make Arab nations leave the coalition.) Keeping the coalition together, especially the Arab nations, was a key priority.

But the coalition did NOT have authority to depose Saddam Hussein. Only to force him from Kuwait. Bush feared that if he overstepped the UN, his coalition would fall apart. (Consider the optics of an "infidel" nation slaughtering Arabs & killing their leader; if you were an Arab dictator & part of this, there was an obvious risk of blowback.)

So, tactically, the US tried to weaken Saddam's military to the point that he could be overthrown internally, encouraging mistreated & abused groups to rebel (such as the Kurds). The US strongly suggested / promised to assist these groups.

After a loooong bombing campaign, the coalition troops had a 100 hour ground war & drove Iraq out of Kuwait. Big success. Then they stopped. Groups like the Kurds then rebelled.

And... They were slaughtered. Saddam's forces were stronger than we anticipated. Turns out, the Kurds needed Coalition / US nilitary support to succeed, and that support never arrived.

So Saddam stayed in power.