After years of kiting both strapped and strapless, here is my take on this endless debate.
There are exactly two scenarios in which strapless kiting is better than strapped kiting:
1) The Wind is cross off, so the water is smooth
2) The wind is light and you don't have a foil

IMO, there are no other scenarios in which strapless kiting is better than strapped kiting.

Since most places don't offer safe cross-off wave locations, this only leaves scenario 2.
Onshore wind directions always means the water is choppy as feck. Chop sucks bad on strapless. But before I don't kite at all in light wind, I take my surfboard. And that's the only reason I haven't gotten rid of it.

For any other wave scenario, strapped kiting is so much better. I can attack huge sets very aggressively, whack the tail around, boost over waves and have a huge grin on my face. I'm sure if you're an absolute pro, you can do all that with a strapless board as well. But if you're not, then you can't.

And I'm not worried about not looking cool with my straps... retrieving the strapless board in huge surf every 5 minutes...very uncool.

Disclaimer: This is purely based on my personal experience. But feel free to rip into me if you think I'm full of it.