Great video, and regardless of board stability, it's our challenge to stay upright.
Some things I've learned from surfing a beach break all the time, where constantly faced with whitewater:
Keep paddling through white water .. many just stop paddling and lose the stability / momentum. Having your paddle in the water helps plenty.
If it's powerful white water, it's more stable to quickly switch to a wide surf stance, get low, punch through, then switch back.
Depending on the board/wave, you can choose to punch the nose through the wave, rather than ride over the top.
Timing .. read the wave, and paddle like buggery if it looks like it will break, versus, holding back, letting it break/avoid the impact
If there's a peak, and you can make it, paddle sideways as dogman in the video did to completely avoid the whitewater.
On bigger days, best option can be to drop down with paddle under chest, and prone paddle out, as getting up/down wastes the precious seconds you have to get out between sets.
If you're going to take one on the head, then do a dive off the front of your board, rather than just fall in, as every meter gained towards getting out the back helps.
Conserve or Go like hell? Depends on the break, but sometimes, when it's big, and waves are coming through fast, then get a move on! Getting out the back as fast as possible will save time & conserve energy. Other days, it can pay to read the rips, and be more relaxed, take a few on the head, and time your runs to get out the back.
It's all good fun...